A First Course in Linear Algebra

by Rob Beezer

 

1 Systems of Linear Equations 

WILA What is Linear Algebra 

LA Linear + Algebra 


We begin our study of linear algebra with an introduction and a motivational example.

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AA An Application 

SSLE Solving Systems of Linear Equations 

SLE Systems of Linear Equations 


We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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 3 years ago
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We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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PSS Possibilities for Solution Sets 

ESEO Equivalent Systems and Equation Operations 


Equivalence of systems of linear equations, row operations, corresponding matrices representing the linear systems

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We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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September 11th, 2017
 3 years ago
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 English
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We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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September 11th, 2017
 3 years ago
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We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

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 3 years ago
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RREF Reduced Row-Echelon Form 

MVNSE Matrix and Vector Notation for Systems of Equations 


Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

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3 years ago
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Definition of vector, equality of vectors, vector addition, and scalar vector multiplication. Geometric and algebraic properties of vector addition are discussed. (need a topic on vector addition is commutative and associative)

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University of Waterloo Math Online -

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Slides for the accompanying video from University of Waterloo.

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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

  • math.la.d.mat.constant
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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 3 years ago
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

License
GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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 3 years ago
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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 3 years ago
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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 3 years ago
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 English
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RO Row Operations 


Equivalence of systems of linear equations, row operations, corresponding matrices representing the linear systems

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3 years ago
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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GFDL-1.2
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 3 years ago
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2
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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 3 years ago
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math.la.d.mat.row_op .d.mat.row_equiv

RREF Reduced Row-Echelon Form 


A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

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February 15th, 2017
3 years ago
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Definition of echelon form, reduction of a matrix to echelon form in order to compute solutions to systems of linear equations; definition of reduced row echelon form

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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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GFDL-1.2
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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GFDL-1.2
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 3 years ago
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After solving a few systems of equations, you will recognize that it does not matter so much what we call our variables, as opposed to what numbers act as their coefficients. A system in the variables \(x_1,\,x_2,\,x_3\) would behave the same if we changed the names of the variables to \(a,\,b,\,c\) and kept all the constants the same and in the same places. In this section, we will isolate the key bits of information about a system of equations into something called a matrix, and then use this matrix to systematically solve the equations. Along the way we will obtain one of our most important and useful computational tools.

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TSS Types of Solution Sets 

CS Consistent Systems 


A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

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February 15th, 2017
3 years ago
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Sample problems to help understand when a linear system has 0, 1, or infinitely many solutions.

  • Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi
  • math.la.t.rref.consistent
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

  • math.la.t.rref.pivot
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

  • math.la.t.rref.consistent
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

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 3 years ago
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FV Free Variables 


A 3x3 system having a unique solution is solved by putting the augmented matrix in reduced row echelon form. A picture of three intersecting planes provides geometric intuition.

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3 years ago
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Sample problems to help understand when a linear system has 0, 1, or infinitely many solutions.

  • Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi
  • math.la.t.rref.consistent
Created On
February 15th, 2017
3 years ago
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

  • The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set. math.la.t.rref.pivot.free
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We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

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 3 years ago
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HSE Homogeneous Systems of Equations 

SHS Solutions of Homogeneous Systems 


How to compute all solutions to a general system $Ax=b$ of linear equations and connection to the corresponding homogeneous system $Ax=0$. Visualization of the geometry of solution sets. Consistent systems and their solution using row reduction.

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August 22nd, 2017
3 years ago
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Homogeneous systems of linear equations; trivial versus nontrivial solutions of homogeneous systems; how to find nontrivial solutions; how to know from the reduced row-echelon form of a matrix whether the corresponding homogeneous system has nontrivial solutions.

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August 25th, 2017
3 years ago
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In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

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In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

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 3 years ago
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In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

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 3 years ago
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In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

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 3 years ago
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NSM Null Space of a Matrix 


In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

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NM Nonsingular Matrices 

NM Nonsingular Matrices 


Motivation and definition of the inverse of a matrix

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(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
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3 years ago
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Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

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3 years ago
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The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

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February 17th, 2017
3 years ago
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Statements that are equivalent to a square matrix being invertible; examples.

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August 21st, 2017
3 years ago
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Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

Created On
August 25th, 2017
3 years ago
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 English
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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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 3 years ago
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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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 3 years ago
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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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NSNM Null Space of a Nonsingular Matrix 


In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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 3 years ago
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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

  • math.la.t.equiv.identity.rep
  • math.la.t.equiv.nullspace.rep
  • math.la.t.equiv.mat.eqn.unique.rep
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 3 years ago
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In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (Chapter D, Chapter E, Chapter LT, Chapter R) that these matrices are especially important.

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 3 years ago
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2 Vectors 

VO Vector Operations 

CV Column Vectors 


Definition of vector, equality of vectors, vector addition, and scalar vector multiplication. Geometric and algebraic properties of vector addition are discussed. (need a topic on vector addition is commutative and associative)

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University of Waterloo Math Online -

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Slides for the accompanying video from University of Waterloo.

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October 23rd, 2013
7 years ago
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Quiz from the University of Waterloo.

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October 23rd, 2013
7 years ago
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In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

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 3 years ago
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In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

  • math.la.d.vec.rncn
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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VSP Vector Space Properties 


In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.

  • math.la.t.vec.sum.coord
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 3 years ago
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LC Linear Combinations 

LC Linear Combinations 


Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

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February 15th, 2017
3 years ago
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A 3x3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions. These examples illustrate a theorem about linear combinations of the columns of the matrix A.

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February 15th, 2017
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The linear combination of a set of vectors is defined. Determine if a vector in R^2 is in the span of two other vectors. The span of a set of vectors is related to the columns of a matrix. (need topic: Determine if a vector in R^2 is in the span of two other vectors.)

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February 20th, 2017
3 years ago
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Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

Created On
February 20th, 2017
3 years ago
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Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

Created On
February 20th, 2017
3 years ago
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2
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 In-class
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 English
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In-class activity for linear combinations and span.

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GFDL-1.3
Created On
June 8th, 2017
3 years ago
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3
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 In-class
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 Introduction
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University of Waterloo Math Online -

Created On
October 23rd, 2013
7 years ago
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Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

Created On
August 25th, 2017
3 years ago
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 Video
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In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

  • The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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 3 years ago
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In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

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GFDL-1.2
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 3 years ago
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A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

  • The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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 3 years ago
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VFSS Vector Form of Solution Sets 


In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

  • math.la.t.linsys.soln.vector
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 3 years ago
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PSHS Particular Solutions, Homogeneous Solutions 


In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

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GFDL-1.2
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September 11th, 2017
 3 years ago
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SS Spanning Sets 

SSV Span of a Set of Vectors 


Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

Created On
February 20th, 2017
3 years ago
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3
Type
 Video
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 Introduction
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 English
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Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

Created On
February 20th, 2017
3 years ago
Views
2
Type
 Handout
Timeframe
 In-class
Language
 English
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text/html; charset=utf-8

In-class activity for linear combinations and span.

License
GFDL-1.3
Created On
June 8th, 2017
3 years ago
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3
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From the University of Waterloo Math Online

Created On
October 23rd, 2013
7 years ago
Views
3
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 Video
Perspective
 Introduction
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Slides from the corresponding video from the University of Waterloo.

Created On
October 23rd, 2013
7 years ago
Views
3
Type
 Handout
Perspective
 Introduction
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 English
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application/pdf

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

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September 11th, 2017
 3 years ago
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 Textbook
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SSV Span of a Set of Vectors 


Definition of the span of a set of vectors. Example of checking if a vector in R^3 is in the span of a set of two vectors. Geometric picture of a span.

Created On
February 20th, 2017
3 years ago
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3
Type
 Video
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 Pre-class
Perspective
 Introduction
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 English
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Suggestions for in-class activities on linear combination and span of vectors in R^n. (need a topic for the general *process* of determining if a vector is in the span of a set of devtors)

Created On
February 20th, 2017
3 years ago
Views
2
Type
 Handout
Timeframe
 In-class
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 English
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text/html; charset=utf-8

In-class activity for linear combinations and span.

License
GFDL-1.3
Created On
June 8th, 2017
3 years ago
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3
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 Handout
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 In-class
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 Introduction
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From the University of Waterloo Math Online

Created On
October 23rd, 2013
7 years ago
Views
3
Type
 Video
Perspective
 Introduction
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 English
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Slides from the corresponding video from the University of Waterloo.

Created On
October 23rd, 2013
7 years ago
Views
3
Type
 Handout
Perspective
 Introduction
Language
 English
Content Type
application/pdf

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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SSNS Spanning Sets of Null Spaces 


In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

  • math.la.t.mat.null_space.rref.span
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 3 years ago
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LI Linear Independence 

LISV Linearly Independent Sets of Vectors 


Linear independence is defined, followed by a worked example of 3 vectors in R^3.

  • Determine if a particular set of vectors in R^3 in linearly independent math.la.e.vec.linindep.r3
  • Definition of linearly independent set of vectors: if a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.d.vec.linindep.coord
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Linear independence in-class activity

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3 years ago
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Video Lesson from University of Waterloo.

Created On
October 23rd, 2013
7 years ago
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Quiz from the University of Waterloo.

Created On
October 23rd, 2013
7 years ago
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

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GFDL-1.2
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September 11th, 2017
 3 years ago
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

  • Definition of linearly independent set of vectors: if a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.d.vec.linindep.coord
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

  • math.la.t.vec.linindep.pivot
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 3 years ago
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

  • math.la.t.vec.linindep.homog
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GFDL-1.2
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 3 years ago
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LINM Linear Independence and Nonsingular Matrices 


Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

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 3 years ago
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Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

  • math.la.t.equiv.col.linindep.rep
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 3 years ago
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NSSLI Null Spaces, Spans, Linear Independence 


Linear independence is one of the most fundamental conceptual ideas in linear algebra, along with the notion of a span. So this section, and the subsequent Section LDS, will explore this new idea.

  • math.la.t.mat.null_space.rref.basis
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 3 years ago
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LDS Linear Dependence and Spans 

LDSS Linearly Dependent Sets and Spans 


Linear independence in-class activity

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Video Lesson from University of Waterloo.

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October 23rd, 2013
7 years ago
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Quiz from the University of Waterloo.

Created On
October 23rd, 2013
7 years ago
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In any linearly dependent set there is always one vector that can be written as a linear combination of the others. This is the substance of the upcoming Theorem DLDS. Perhaps this will explain the use of the word “dependent.” In a linearly dependent set, at least one vector “depends” on the others (via a linear combination).

  • Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
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 3 years ago
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COV Casting Out Vectors 


In any linearly dependent set there is always one vector that can be written as a linear combination of the others. This is the substance of the upcoming Theorem DLDS. Perhaps this will explain the use of the word “dependent.” In a linearly dependent set, at least one vector “depends” on the others (via a linear combination).

  • math.la.t.vsp.span.basis.rref
  • A set of nonzero vectors contains (as a subset) a basis for its span. math.la.t.vsp.span.basis
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O Orthogonality 

CAV Complex Arithmetic and Vectors 


In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.d.vec.conjugate.cn
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 3 years ago
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.vec.sum.conjugate.cn
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GFDL-1.2
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September 11th, 2017
 3 years ago
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.vec.scalar.mult.conjugate.cn
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GFDL-1.2
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September 11th, 2017
 3 years ago
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IP Inner products 


In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.innerproduct.commutative.scalar.cn
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 3 years ago
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.innerproduct.distributive.rncn
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.innerproduct.commutative.cn
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
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 English
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.d.innerproduct.cn
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GFDL-1.2
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September 11th, 2017
 3 years ago
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N Norm 


This is a video from the University of Waterloo. Dot Product, Cross-Product in R^n (which should be in Chapter 8 section 4 about hyperplanes.

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October 23rd, 2013
7 years ago
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Quiz from the University of Waterloo. This is intended to be used after the video of the same name.

Created On
October 23rd, 2013
7 years ago
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Inner product of two vectors in R^n, length of a vector in R^n, orthogonality. Motivation via approximate solutions of systems of linear equations, definition and properties of inner product (symmetric, bilinar, positive definite); length/norm of a vector, unit vectors; definition of distance between vectors; definition of orthogonality; Pythagorean Theorem.

Created On
August 22nd, 2017
3 years ago
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2
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 Video
Language
 English
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In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.t.vec.innerproduct.norm
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

OV Orthogonal Vectors 


This is a video from the University of Waterloo. Dot Product, Cross-Product in R^n (which should be in Chapter 8 section 4 about hyperplanes.

Created On
October 23rd, 2013
7 years ago
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3
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 Video
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 Introduction
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 English
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Quiz from the University of Waterloo. This is intended to be used after the video of the same name.

Created On
October 23rd, 2013
7 years ago
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3
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 Unknown
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 Post-class
Perspective
 Example
Language
 English
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text/html;charset=UTF-8

Inner product of two vectors in R^n, length of a vector in R^n, orthogonality. Motivation via approximate solutions of systems of linear equations, definition and properties of inner product (symmetric, bilinar, positive definite); length/norm of a vector, unit vectors; definition of distance between vectors; definition of orthogonality; Pythagorean Theorem.

Created On
August 22nd, 2017
3 years ago
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2
Type
 Video
Language
 English
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text/html; charset=utf-8

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

  • math.la.d.vec.orthogonal.coord
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

GSP Gram-Schmidt Procedure 


Orthonormal sets and bases (definition); expressing vectors as linear combinations of orthonormal basis vectors; matrices with orthonormal columns preserve vector norm and dot product; orthogonal matrices; inverse of an orthogonal matrix equals its transpose

Created On
August 25th, 2017
3 years ago
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3
Type
 Video
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 English
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text/html; charset=utf-8

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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text/html

In this section we define a couple more operations with vectors, and prove a few theorems. At first blush these definitions and results will not appear central to what follows, but we will make use of them at key points in the remainder of the course (such as Section MINM, Section OD). Because we have chosen to use \(\complexes\) as our set of scalars, this subsection is a bit more, uh, … complex than it would be for the real numbers. We will explain as we go along how things get easier for the real numbers \({\mathbb R}\text{.}\) If you have not already, now would be a good time to review some of the basic properties of arithmetic with complex numbers described in Section CNO. With that done, we can extend the basics of complex number arithmetic to our study of vectors in \(\complex{m}\text{.}\)

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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

3 Matrices 

MO Matrix Operations 

MEASM Matrix Equality, Addition, Scalar Multiplication 


The product of a matrix times a vector is defined, and used to show that a system of linear equations is equivalent to a system of linear equations involving matrices and vectors. The example uses a 2x3 system.

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CC-BY-SA-4.0
Created On
February 15th, 2017
3 years ago
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3
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

Definition of sum of matrices, product of a scalar and a matrix

Created On
February 17th, 2017
3 years ago
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3
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 Video
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 Pre-class
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 Introduction
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 English
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text/html; charset=utf-8

Suggestions for in-class activities on matrix operations: addition, multiplication, transpose, and the fact that multiplication is not commutative.

Created On
February 17th, 2017
3 years ago
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2
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 Handout
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 In-class
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 English
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In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.d.mat.m_by_n.set
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

VSP Vector Space Properties 


In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.d.mat.z
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.e.vsp.mat.m_by_n
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

  • math.la.e.vsp.mat.m_by_n
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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TSM Transposes and Symmetric Matrices 


The transpose of a matrix is defined, and various properties are explored using numerical examples.

Created On
February 17th, 2017
3 years ago
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2
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

Associative and distributive properties of matrix multiplication and addition; multiplication by the identity matrix; definition of the transpose of a matrix; transpose of the transpose, transpose of a sum, transpose of a product

Created On
August 25th, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.sum.transpose
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.symmetric.square
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.scalar.transpose
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

MCC Matrices and Complex Conjugation 


In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.scalar.conjugate
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.conjugate.involution
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.d.mat.conjugate
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.matsum.conjugate
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.transpose.conjugate
License
GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

AM Adjoint of a Matrix 


In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.d.mat.adjoint
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.adjoint.involution
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.d.mat.skewsymmetric
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.sum.adjoint
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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text/html

In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.t.mat.scalar.adjoint
License
GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

MM Matrix Multiplication 

MVP Matrix-Vector Product 


The product of a matrix times a vector is defined, and used to show that a system of linear equations is equivalent to a system of linear equations involving matrices and vectors. The example uses a 2x3 system.

License
CC-BY-SA-4.0
Created On
February 15th, 2017
3 years ago
Views
3
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

Created On
February 15th, 2017
3 years ago
Views
3
Type
 Handout
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 In-class
Perspective
 Example
Language
 English
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text/html; charset=utf-8

Learning goals: 1. What are the dimension (size) requirements for two matrices so that they can be multiplied to each other? 2. What is the product of two matrices, when it exists?

Created On
February 17th, 2017
3 years ago
Views
2
Type
 Video
Timeframe
 Review
Language
 English
Content Type
text/html; charset=utf-8

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.mat.vec.prod.unique
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

MM Matrix Multiplication 


Learning goals: 1. What are the dimension (size) requirements for two matrices so that they can be multiplied to each other? 2. What is the product of two matrices, when it exists?

Created On
February 17th, 2017
3 years ago
Views
2
Type
 Video
Timeframe
 Review
Language
 English
Content Type
text/html; charset=utf-8

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.c.mat.mult
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

MMEE Matrix Multiplication, Entry-by-Entry 


We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.mat.mult.coord
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

PMM Properties of Matrix Multiplication 


The transpose of a matrix is defined, and various properties are explored using numerical examples.

Created On
February 17th, 2017
3 years ago
Views
2
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

Created On
February 17th, 2017
3 years ago
Views
2
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

Associative and distributive properties of matrix multiplication and addition; multiplication by the identity matrix; definition of the transpose of a matrix; transpose of the transpose, transpose of a sum, transpose of a product

Created On
August 25th, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.mat.mult.conjugate
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.mat.mult.z
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
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 Textbook
Language
 English
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text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.innerproduct.mat.cn
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.mat.mult.adjoint
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

HM Hermitian Matrices 


We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.t.innerproduct.adjoint.cn
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We know how to add vectors and how to multiply them by scalars. Together, these operations give us the possibility of making linear combinations. Similarly, we know how to add matrices and how to multiply matrices by scalars. In this section we mix all these ideas together and produce an operation known as matrix multiplication. This will lead to some results that are both surprising and central. We begin with a definition of how to multiply a vector by a matrix.

  • math.la.d.mat.hermitian.innerproduct.cn
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

MISLE Matrix Inverses and Systems of Linear Equations 

SI Solutions and Inverses 

IM Inverse of a Matrix 


Motivation and definition of the inverse of a matrix

License
(CC-BY-NC-SA-4.0 OR CC-BY-SA-4.0)
Created On
January 5th, 2017
3 years ago
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3
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 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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The definition of matrix inverse is motivated by considering multiplicative inverse. The identity matrix and matrix inverse are defined.

Created On
February 17th, 2017
3 years ago
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2
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 Video
Timeframe
 Pre-class
Perspective
 Introduction
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 English
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Suggested classroom activities on matrix inverses.

Created On
February 19th, 2017
3 years ago
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2
Type
 Handout
Timeframe
 In-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

Created On
June 8th, 2017
3 years ago
Views
2
Type
 Handout
Perspective
 Application
Language
 English
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text/html; charset=utf-8

Statements that are equivalent to a square matrix being invertible; examples.

Created On
August 21st, 2017
3 years ago
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3
Type
 Video
Language
 English
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text/html; charset=utf-8

Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

Created On
August 22nd, 2017
3 years ago
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3
Type
 Video
Language
 English
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text/html; charset=utf-8

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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CIM Computing the Inverse of a Matrix 


Matrix inverses are motivated as a way to solve a linear system. The general algorithm of finding an inverse by row reducing an augmented matrix is described, and then implemented for a 3x3 matrix. Useful facts about inverses are stated and then illustrated with sample 2x2 matrices. (put first: need Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix)

Created On
February 19th, 2017
3 years ago
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3
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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Suggested classroom activities on matrix inverses.

Created On
February 19th, 2017
3 years ago
Views
2
Type
 Handout
Timeframe
 In-class
Perspective
 Introduction
Language
 English
Content Type
text/html; charset=utf-8

Statements that are equivalent to a square matrix being invertible; examples.

Created On
August 21st, 2017
3 years ago
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3
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 Video
Language
 English
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text/html; charset=utf-8

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

Created On
August 25th, 2017
3 years ago
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3
Type
 Video
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 English
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text/html; charset=utf-8

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

  • The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix. math.la.t.mat.inv.augmented
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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PMI Properties of Matrix Inverses 


Statements that are equivalent to a square matrix being invertible; examples.

Created On
August 21st, 2017
3 years ago
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3
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 Video
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 English
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text/html; charset=utf-8

Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

Created On
August 25th, 2017
3 years ago
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3
Type
 Video
Language
 English
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text/html; charset=utf-8

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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text/html

The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
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 English
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The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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The inverse of a square matrix, and solutions to linear systems with square coefficient matrices, are intimately connected.

  • math.la.t.mat.inv.scalar
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 English
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MINM Matrix Inverses and Nonsingular Matrices 

NMI Nonsingular Matrices are Invertible 


Matrix inverses are motivated as a way to solve a linear system. The general algorithm of finding an inverse by row reducing an augmented matrix is described, and then implemented for a 3x3 matrix. Useful facts about inverses are stated and then illustrated with sample 2x2 matrices. (put first: need Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix)

Created On
February 19th, 2017
3 years ago
Views
3
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

Created On
June 8th, 2017
3 years ago
Views
2
Type
 Handout
Perspective
 Application
Language
 English
Content Type
text/html; charset=utf-8

Statements that are equivalent to a square matrix being invertible; examples.

Created On
August 21st, 2017
3 years ago
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3
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

Created On
August 22nd, 2017
3 years ago
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3
Type
 Video
Language
 English
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text/html; charset=utf-8

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

  • math.la.t.mat.prod.nonsingular
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

  • math.la.t.mat.inv.oneside
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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UM Unitary Matrices 


We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

  • math.la.t.mat.unitary.col.orthogonal
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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text/html

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

  • math.la.t.mat.unitary.innerproduct
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

  • math.la.t.mat.unitary.inv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

We saw in Theorem CINM that if a square matrix \(A\) is nonsingular, then there is a matrix \(B\) so that \(AB=I_n\text{.}\) In other words, \(B\) is halfway to being an inverse of \(A\text{.}\) We will see in this section that \(B\) automatically fulfills the second condition (\(BA=I_n\)). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
Language
 English
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text/html

CRS Column and Row Spaces 

CSSE Column Spaces and Systems of Equations 


Advice to instructors for in-class activities on matrix-vector multiplication and translating between the various equivalent notation forms of linear systems, and suggestions for how this topic can be used to motivate future topics.

Created On
February 15th, 2017
3 years ago
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3
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 Handout
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 In-class
Perspective
 Example
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 English
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text/html; charset=utf-8

A 3x3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions. These examples illustrate a theorem about linear combinations of the columns of the matrix A.

Created On
February 15th, 2017
3 years ago
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3
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 Video
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 Pre-class
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 Example
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 English
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The linear combination of a set of vectors is defined. Determine if a vector in R^2 is in the span of two other vectors. The span of a set of vectors is related to the columns of a matrix. (need topic: Determine if a vector in R^2 is in the span of two other vectors.)

Created On
February 20th, 2017
3 years ago
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2
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 Video
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 Pre-class
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 Introduction
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 English
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Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

Created On
August 25th, 2017
3 years ago
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2
Type
 Video
Language
 English
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text/html; charset=utf-8

Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

Created On
August 25th, 2017
3 years ago
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3
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 Video
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 English
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text/html; charset=utf-8

In Section VO we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

  • The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

  • The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

License
GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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CSSOC Column Space Spanned by Original Columns 


Basis theorem: for an n-dimensional vector space any linearly independent set with n elements is a basis, as is any spanning set with n elements; dimension of the column space of a matrix equals the number of pivot columns of the matrix; dimension of the null space of a matrix equals the number of free variables of the matrix

Created On
August 25th, 2017
3 years ago
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3
Type
 Video
Language
 English
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text/html; charset=utf-8

The pivot columns of a matrix form a basis for its column space; nullspace of a matrix equals the nullspace of its reduced row-echelon form.

Created On
August 25th, 2017
3 years ago
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3
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 Video
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 English
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text/html; charset=utf-8

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

License
GFDL-1.2
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September 11th, 2017
 3 years ago
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4
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 Textbook
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 English
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text/html

CSNM Column Space of a Nonsingular Matrix 


Equivalent statements for a matrix A: for every right-hand side b, the system Ax=b has a solution; every b is a linear combination of the columns of A; the span of the columns of A is maximal; A has a pivot position in every row.

Created On
August 25th, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

RSM Row Space of a Matrix 


A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

  • math.la.t.mat.row_space.pivot
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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

A matrix-vector product (Definition MVP) is a linear combination of the columns of the matrix and this allows us to connect matrix multiplication with systems of equations via Theorem SLSLC. Row operations are linear combinations of the rows of a matrix, and of course, reduced row-echelon form (Definition RREF) is also intimately related to solving systems of equations. In this section we will formalize these ideas with two key definitions of sets of vectors derived from a matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

FS Four Subsets 

LNS Left Null Space 


There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

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GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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CCS Computing Column Spaces 

EEF Extended Echelon Form 


There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

  • math.la.t.mat.erref.of
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

  • math.la.d.mat.erref.of
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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text/html

FS Four Subsets 


There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

  • math.la.c.mat.col_space.row_reduce
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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There are four natural subsets associated with a matrix. We have met three already: the null space, the column space and the row space. In this section we will introduce a fourth, the left null space. The objective of this section is to describe one procedure that will allow us to find linearly independent sets that span each of these four sets of column vectors. Along the way, we will make a connection with the inverse of a matrix, so Theorem FS will tie together most all of this chapter (and the entire course so far).

  • math.la.t.mat.erref.spaces
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4 Vector Spaces 

VS Vector Spaces 

VS Vector Spaces 


This video kicks off the series of videos on vector spaces. We begin by summarizing the essential properties of R^n.

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In this video we continue to list the properties of R^n. The 10 properties listed in this video and the previous video will be used to define a general vecto...

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The concept of a vector space is somewhat abstract, and under this definition, a lot of objects such as polynomials, functions, etc., can be considered as vectors. This video explains the definition of a general vector space. In later videos we will look at more examples.

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Definition of a (real) vector space; properties of the zero vector and the additive inverse in relation to scalar multiplication

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In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.

  • math.la.e.vsp.mat.m_by_n
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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

  • math.la.e.vsp.polynomial.leq_n
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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

  • math.la.e.vsp.mat.m_by_n
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VSP Vector Space Properties 


Definition of a (real) vector space; properties of the zero vector and the additive inverse in relation to scalar multiplication

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

  • math.la.t.vsp.z.unique
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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

  • math.la.d.vsp.vector.negative.unique
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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

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RD Recycling Definitions 

S Subspaces 

S Subspaces 


Preliminaries: 1. What is a subset? 2. How to verify a set is a subset of another set? 3. Notations and language of set theory related to subsets. In this video, we introduce the definition of a subspace. We go through a preliminary example to figure out what do subspaces of R^2 look like, and we will continue to talk about how to verify a subset of a vector space is a subspace in later videos.

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In this video, I'll explain why we only need to test 2 axioms (among the 10 axioms in the definition of a vector space) when figuring out if a subset is a subspace.

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Definition of a subspace of a vector space; examples; span of vectors is a subspace.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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TS Testing Subspaces 


A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

  • math.la.t.vsp.subspace.lincomb.arb
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TSS The Span of a Set 


Definition of a vector; vector addition; scalar multiplication; visualization in R^2 and R^3; vector space axioms; linear combinations; span.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

  • math.la.t.vec.span.subspace.arb
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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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SC Subspace Constructions 


A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

  • math.la.t.mat.null_space.left.rncn
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LISS Linear Independence and Spanning Sets 

LI Linear Independence 


A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

  • Definition of linearly indepentent set of vectors: if a linear combination is zero, then every coefficient is zero, arbitrary vector space. math.la.d.vec.linindep.arb
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A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

  • math.la.d.vec.lindep.relation.rep
  • math.la.d.vec.lindep.relation.trvial.rep
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SS Spanning Sets 


A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

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VR Vector Representation 


Representation (unique) of a vector in terms of a basis for a vector space yields coordinates relative to the basis; change of basis and corresponding change of coordinate matrix

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A vector space is defined as a set with two operations, meeting ten properties (Definition VS). Just as the definition of span of a set of vectors only required knowing how to add vectors and how to multiply vectors by scalars, so it is with linear independence. A definition of a linearly independent set of vectors in an arbitrary vector space only requires knowing how to form linear combinations and equating these with the zero vector. Since every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal.

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B Bases 

B Bases 


A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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BSCV Bases for Spans of Column Vectors 

BNM Bases and Nonsingular Matrices 


A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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OBC Orthonormal Bases and Coordinates 


A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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A basis of a vector space is one of the most useful concepts in linear algebra. It often provides a concise, finite description of an infinite vector space.

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D Dimension 

D Dimension 


Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

  • A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space. math.la.t.vsp.dim.more.lindep.arb
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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

  • Every basis for a vector space contains the same number of elements, arbitrary vector space. math.la.t.vsp.dim.arb
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DVS Dimension of Vector Spaces 

RNM Rank and Nullity of a Matrix 


Students answer multiple questions on the rank and dimension of the null space in a variety of situations to discover the connection between these dimensions leading to the Rank-Nullity Theorem.

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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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 3 years ago
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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

  • math.la.t.mat.rank.pivot
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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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 3 years ago
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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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 3 years ago
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RNNM Rank and Nullity of a Nonsingular Matrix 


Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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 3 years ago
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PD Properties of Dimension 

GT Goldilocks' Theorem 


Basis theorem: for an n-dimensional vector space any linearly independent set with n elements is a basis, as is any spanning set with n elements; dimension of the column space of a matrix equals the number of pivot columns of the matrix; dimension of the null space of a matrix equals the number of free variables of the matrix

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Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

  • math.la.t.vsp.linindep.extend
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 3 years ago
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Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

  • A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space. math.la.t.vsp.dim.more.lindep.arb
  • math.la.t.vsp.dim.less.span.arb
  • math.la.t.vsp.dim.span.linindep.arb
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 3 years ago
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Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

  • math.la.t.vsp.subspace.dim.equal
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September 11th, 2017
 3 years ago
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Almost every vector space we have encountered has been infinite in size (an exception is Example VSS). But some are bigger and richer than others. Dimension, once suitably defined, will be a measure of the size of a vector space, and a useful tool for studying its properties. You probably already have a rough notion of what a mathematical definition of dimension might be — try to forget these imprecise ideas and go with the new ones given here.

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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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RT Ranks and Transposes 


Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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DFS Dimension of Four Subspaces 


Once the dimension of a vector space is known, then the determination of whether or not a set of vectors is linearly independent, or if it spans the vector space, can often be much easier. In this section we will state a workhorse theorem and then apply it to the column space and row space of a matrix. It will also help us describe a super-basis for \(\complex{m}\text{.}\)

  • math.la.t.mat.erref.dimension
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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5 Determinants 

DM Determinant of a Matrix 

EM Elementary Matrices 


Properties of matrix inversion: inverse of the inverse, inverse of the transpose, inverse of a product; elementary matrices and corresponding row operations; a matrix is invertible if and only if it is row-equivalent to the identity matrix; row-reduction algorithm for computing matrix inverse

Created On
August 25th, 2017
3 years ago
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3
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 Video
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

  • math.la.d.mat.elementary.prod
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

  • math.la.t.mat.mult.elementary
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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DD Definition of the Determinant 


The formula for the inverse of a 2x2 matrix is derived. (need tag for that formula)

Created On
February 17th, 2017
3 years ago
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2
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 Video
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 Pre-class
Perspective
 Introduction
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 English
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Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.

Created On
August 22nd, 2017
3 years ago
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3
Type
 Video
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 English
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text/html; charset=utf-8

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

  • math.la.c.mat.det
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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CD Computing Determinants 


Determinant of the transpose equals the determinant of the original matrix; rescaling a column rescales the determinant by the same factor; interchanging two columns changes the sign of the determinant; adding multiple of one column to another leaves determinant unchanged; determinant of the product of two matrices equals product of the two determinants

Created On
August 25th, 2017
3 years ago
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3
Type
 Video
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

  • math.la.t.mat.det.cofactor.row
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
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 English
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Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

  • math.la.t.mat.det.cofactor.col
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

PDM Properties of Determinants of Matrices 

DRO Determinants and Row Operations 


We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

  • math.la.t.mat.det.add_mult
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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text/html

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

  • math.la.t.mat.det.scalar
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

  • math.la.t.mat.row.z
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

  • math.la.t.mat.det.switch
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

  • math.la.t.mat.row.equal
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

DROEM Determinants, Row Operations, Elementary Matrices 


We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

DNMMM Determinants, Nonsingular Matrices, Matrix Multiplication 


The effect of row operations on the determinant of a matrix; computing determinants via row reduction; a square matrix is invertible if and only if its determinant is nonzero.

Created On
August 22nd, 2017
3 years ago
Views
3
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

Determinant of the transpose equals the determinant of the original matrix; rescaling a column rescales the determinant by the same factor; interchanging two columns changes the sign of the determinant; adding multiple of one column to another leaves determinant unchanged; determinant of the product of two matrices equals product of the two determinants

Created On
August 25th, 2017
3 years ago
Views
3
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in Theorem SMZD and Theorem DRMM, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

6 Eigenvalues 

EE Eigenvalues and Eigenvectors 

EEM Eigenvalues and Eigenvectors of a Matrix 


An introductory activity on eigenvalues and eigenvectors in which students do basic matrix-vector multiplication calculations to find whether given vectors are eigenvectors, to determine the eigenvalue corresponding to an eigenvector and to find an eigenvector corresponding to an eigenvalue. This activity is self-contained and does not require any previous experience with eigenvalues or eigenvectors.

Created On
June 9th, 2017
3 years ago
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2
Type
 Handout
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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PM Polynomials and Matrices 

EEE Existence of Eigenvalues and Eigenvectors 

CEE Computing Eigenvalues and Eigenvectors 


Definition of the eigenspace corresponding to an eigenvector $\lambda$ (and proof that this is a vector space); analysis of simple matrices in R^2 and R^3 to visualize the "geometry" of eigenspaces; proof that eigenvectors corresponding to distinct eigenvectors are linearly independent

Created On
August 25th, 2017
3 years ago
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2
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 Video
Language
 English
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text/html; charset=utf-8

Theorem: \lambda is an eigenvalue of a matrix A if and only if \lambda satisfies the characteristic equation det(A-\lambda I) = 0; examples; eigenvalues of triangular matrices are the diagonal entries.

Created On
September 3rd, 2017
3 years ago
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2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

  • math.la.t.mat.eigsp.subsp
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
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 English
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text/html

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

  • math.la.t.mat.eigsp.nullspace
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
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text/html

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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ECEE Examples of Computing Eigenvalues and Eigenvectors 


In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

  • math.la.d.mat.eig.multiplicity.geometric
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

  • math.la.d.mat.eig.multiplicity.algebraic
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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text/html

PEE Properties of Eigenvalues and Eigenvectors 

BPE Basic Properties of Eigenvalues 


Definition of the eigenspace corresponding to an eigenvector $\lambda$ (and proof that this is a vector space); analysis of simple matrices in R^2 and R^3 to visualize the "geometry" of eigenspaces; proof that eigenvectors corresponding to distinct eigenvectors are linearly independent

Created On
August 25th, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.eig.inv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.eig.scalar
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.eig.polynomial
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.eig.power
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.eig.transpose
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
Language
 English
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text/html

ME Multiplicities of Eigenvalues 


The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.d.mat.eig.number
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

EHM Eigenvalues of Hermitian Matrices 


The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.hermitian.eigvec.orthogonal
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

  • math.la.t.mat.hermitian.eig.real
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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text/html

SD Similarity and Diagonalization 

SM Similar Matrices 


Definition of similarity for square matrices; similarity is an equivalence relation; similar matrices have the same characteristic polynomial and hence the same eigenvalues, with same multiplicities; definition of multiplicity.

Created On
September 3rd, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
Language
 English
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text/html

PSM Properties of Similar Matrices 


Definition of similarity for square matrices; similarity is an equivalence relation; similar matrices have the same characteristic polynomial and hence the same eigenvalues, with same multiplicities; definition of multiplicity.

Created On
September 3rd, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

  • math.la.t.mat.similar.equiv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
Language
 English
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text/html

D Diagonalization 


Notation for matrix entries, diagonal matrix, square matrix, identity matrix, and zero matrix.

Created On
February 17th, 2017
3 years ago
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3
Type
 Video
Timeframe
 Pre-class
Perspective
 Introduction
Language
 English
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text/html; charset=utf-8

A real matrix $A$ is symmetric if and only if it is orthogonally diagonalizable (i.e. $A = PDP^{-1}$ for an orthogonal matrix $P$.) Proof and examples.

Created On
August 21st, 2017
3 years ago
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3
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

Diagonalization theorem: a nxn matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. If so, the matrix factors as A = PDP^{-1}, where D is diagonal and P is invertible (and its columns are the n linearly independent eigenvectors). Algorithm to diagonalize a matrix.

Created On
August 25th, 2017
3 years ago
Views
2
Type
 Video
Language
 English
Content Type
text/html; charset=utf-8

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

  • An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely, and the dimension of each eigenspace equals the multiplicity of the eigenvalue. math.la.t.mat.diagonalizable.charpoly
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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text/html

This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
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This section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

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FS Fibonacci Sequences 


7 Linear Transformations 

LT Linear Transformations 

LT Linear Transformations 


Two proofs, with discussion, of the fact that an abstract linear transformation maps 0 to 0.

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February 15th, 2017
3 years ago
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Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra...

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May 25th, 2017
3 years ago
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This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

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June 8th, 2017
3 years ago
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Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

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August 25th, 2017
3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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LTC Linear Transformation Cartoons 

MLT Matrices and Linear Transformations 


After watching a video defining linear transformations and giving examples of 2-D transformations, students should be able to answer the questions in this quiz.

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3 years ago
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Use matrix transformations to motivate the concept of linear transformation; examples of matrix transformations

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3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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 3 years ago
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LTLC Linear Transformations and Linear Combinations 


Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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PI Pre-Images 


Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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NLTFO New Linear Transformations From Old 


Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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 3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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 3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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 3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

  • math.la.t.lintrans.composition.arb
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 3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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 3 years ago
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Early in Chapter VS we prefaced the definition of a vector space with the comment that it was “one of the two most important definitions in the entire course.” Here comes the other. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. Here we go.

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ILT Injective Linear Transformations 

ILT Injective Linear Transformations 


Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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EILT Examples of Injective Linear Transformations 

KLT Kernel of a Linear Transformation 


Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

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Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

  • math.la.t.lintrans.equiv.kernel
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Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

  • math.la.t.lintrans.preimage.translation.arb
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 3 years ago
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Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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ILTLI Injective Linear Transformations and Linear Independence 


Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

  • math.la.t.lintrans.equiv.basis
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Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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ILTD Injective Linear Transformations and Dimension 


Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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CILT Composition of Injective Linear Transformations 


Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.

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SLT Surjective Linear Transformations 

SLT Surjective Linear Transformations 


The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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ESLT Examples of Surjective Linear Transformations 

RLT Range of a Linear Transformation 


Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation

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3 years ago
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The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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SSSLT Spanning Sets and Surjective Linear Transformations 


The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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SLTD Surjective Linear Transformations and Dimension 


The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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CSLT Composition of Surjective Linear Transformations` 


The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties.

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IVLT Invertible Linear Transformations 

IVLT Invertible Linear Transformations 


In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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IV Invertibility 


In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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SI Structure and Isomorphism 


Given a basis for a n-dimensional vector space V, the coordinate map is a linear bijection between V and R^n; definition isomorphisms between vector spaces and isomorphic vector spaces.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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RNLT Rank and Nullity of a Linear Transformation 


In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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SLELT Systems of Linear Equations and Linear Transformations 


8 Representations 

VR Vector Representations 

VR Vector Representations 


You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

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You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

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You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

  • math.la.t.vsp.basis.coord.surjective.arb
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You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

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CVS Characterization of Vector Spaces 


You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

  • math.la.t.vsp.isomorphic.dim
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In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

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You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

  • math.la.t.vsp.isomorphic.rncn
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CP Coordinatization Principle 


You may have noticed that many questions about elements of abstract vector spaces eventually become questions about column vectors or systems of equations. Example SM32 would be an example of this. We will make this vague idea more precise in this section.

  • math.la.t.vsp.linindep.coord
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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MR Matrix Representations 

MR Matrix Representations 


We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

  • math.la.t.lintrans.mat.repn.arb
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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text/html

NRFO New Representations from Old 


We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

  • math.la.t.lintrans.mat_repn.composition
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GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

  • math.la.t.lintrans.mat_repn.sum
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

  • math.la.t.lintrans.mat_repn.scalar
License
GFDL-1.2
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September 11th, 2017
 3 years ago
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3
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 Textbook
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 English
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PMR Properties of Matrix Representations 

IVLT Invertible Linear Transformations 


We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

  • math.la.t.lintrans.mat_repn.inv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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CB Change of Basis 

EELT Eigenvalues and Eigenvectors of Linear Transformations 


We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

  • math.la.d.lintrans.eig
  • math.la.d.lintrans.eigvec
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GFDL-1.2
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September 11th, 2017
 3 years ago
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CBM Change-of-Basis Matrix 


This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.

Created On
June 8th, 2017
3 years ago
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 English
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We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

  • math.la.t.vsp.change_of_basis
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

  • math.la.t.vsp.change_of_basis.inv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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MRS Matrix Representations and Similarity 


We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

  • math.la.t.lintrans.mat_repn.eig
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section MR that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

  • math.la.t.vsp.change_of_basis.conjugate
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
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 Textbook
Language
 English
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CELT Computing Eigenvectors of Linear Transformations 

OD Orthonormal Diagonalization 

TM Triangular Matrices 


We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.mat.triangular.prod
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
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text/html

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.mat.triangular.inv
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
Views
3
Type
 Textbook
Language
 English
Content Type
text/html

UTMR Upper Triangular Matrix Representation 


We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.lintrans.mat_repn.triangular
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.mat.triangular.unitary
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
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 Textbook
Language
 English
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NM Normal Matrices 


We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.d.mat.normal
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GFDL-1.2
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September 11th, 2017
 3 years ago
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2
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 Textbook
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 English
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OD Orthonormal Diagonalization 


We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.mat.normal.diagonalize
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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2
Type
 Textbook
Language
 English
Content Type
text/html

We have seen in Section SD that under the right conditions a square matrix is similar to a diagonal matrix. We recognize now, via Theorem SCB, that a similarity transformation is a change of basis on a matrix representation. So we can now discuss the choice of a basis used to build a matrix representation, and decide if some bases are better than others for this purpose. This will be the tone of this section. We will also see that every matrix has a reasonably useful matrix representation, and we will discover a new class of diagonalizable linear transformations. First we need some basic facts about triangular matrices.

  • math.la.t.mat.normal.eigenval
License
GFDL-1.2
Submitted At
September 11th, 2017
 3 years ago
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3
Type
 Textbook
Language
 English
Content Type
text/html