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.

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of one solution math.la.e.linsys.3x3.soln.row_reduce.o

##### Definition of matrix in reduced row echelon form math.la.d.mat.rref

##### Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi

##### Definition of consistent linear system math.la.d.linsys.consistent

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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.

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.row_reduce.i

##### 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

##### Definition of matrix equation math.la.d.mat.eqn

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The reduced row echelon form is used to determine when a 3x3 system is inconsistent. A picture of planes in 3-dimensional space is used to provide geometric intuition.

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z

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

##### Notation for entry of matrix math.la.d.mat.entry

##### Definition of m by n matrix math.la.d.mat.m_by_n

##### Definition of diagonal matrix math.la.d.mat.diagonal

##### Definition of identity matrix math.la.d.mat.identity

##### Definition of zero matrix math.la.d.mat.zero

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

##### Definition of equivalent systems of linear equations math.la.d.linsys.equiv

##### Definition of row operations on a matrix math.la.d.mat.row_op

##### Definition of row equivalent matrices math.la.d.mat.row_equiv

##### Row equivalent matrices represent equivalent linear systems math.la.t.mat.row_equiv.linsys

##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of

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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.

##### Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.homog.row_reduce.i

##### Definition of homogeneous linear system of equations math.la.d.linsys.homog

##### A homogeneous system has a nontrivial solution if and only if it has a free variable. math.la.t.linsys.homog.nontrivial

##### Parametric vector form of the solution set of a system of linear equations math.la.c.linsys.soln_set.vec

##### Definition of trivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.trivial

##### Definition of nontrivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.nontrivial

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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.

##### Definition of homogeneous linear system of equations math.la.d.linsys.homog

##### Definition of trivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.trivial

##### Definition of nontrivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.nontrivial

##### A homogeneous system has a nontrivial solution if and only if it has a free variable. math.la.t.linsys.homog.nontrivial

##### Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.homog.row_reduce.i

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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.

##### Definition of system of linear equations math.la.d.linsys

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##### Definition of solution set of a system of linear equations math.la.d.linsys.soln_set

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

##### Definition of linear equation math.la.d.lineqn

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##### Definition of solution to a system of linear equations math.la.d.linsys.soln

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##### Definition of equivalent systems of linear equations math.la.d.linsys.equiv

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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.

##### Definition of matrix math.la.d.mat

##### Definition of m by n matrix math.la.d.mat.m_by_n

##### math.la.c.mat.entry

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##### Definition of row operations on a matrix math.la.d.mat.row_op

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##### Definition of coefficient matrix of a linear system math.la.d.mat.coeff

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##### Definition of augmented matrix of a linear system math.la.d.mat.augmented

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##### Definition of row equivalent matrices math.la.d.mat.row_equiv

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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.

##### Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi

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##### Definition of consistent linear system math.la.d.linsys.consistent

##### Definition of inconsistent linear system math.la.d.linsys.inconsistent

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##### Row equivalent matrices represent equivalent linear systems math.la.t.mat.row_equiv.linsys

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##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of one solution math.la.e.linsys.3x3.soln.row_reduce.o

##### Definition of matrix in reduced row echelon form math.la.d.mat.rref

##### Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi

##### Definition of consistent linear system math.la.d.linsys.consistent

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##### Definition of equivalent systems of linear equations math.la.d.linsys.equiv

##### Definition of row operations on a matrix math.la.d.mat.row_op

##### Definition of row equivalent matrices math.la.d.mat.row_equiv

##### Row equivalent matrices represent equivalent linear systems math.la.t.mat.row_equiv.linsys

##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of

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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

##### Definition of (row) echelon form of a matrix math.la.d.mat.echelon.of

##### Example of (echelon matrix/matrix in (row) echelon form) math.la.e.mat.echelon

##### Definition of matrix in reduced row echelon form math.la.d.mat.rref

##### Definition of reduced row echelon form of a matrix math.la.d.mat.rref.of

##### Definition of leading entry in a row of a matrix math.la.d.mat.row.leading

##### Definition of (echelon matrix/matrix in (row) echelon form) math.la.d.mat.echelon

##### Example of putting a matrix in echelon form math.la.e.mat.echelon.of

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##### Every matrix is row-equivalent to a matrix in reduced row echelon form. math.la.t.mat.rref.exists

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##### Every matrix is row-equivalent to only one matrix in reduced row echelon form. math.la.t.mat.rref.unique

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##### Definition of matrix in reduced row echelon form math.la.d.mat.rref

##### Definition of pivot column math.la.d.mat.pivot_col

##### Definition of leading entry in a row of a matrix math.la.d.mat.row.leading

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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.

##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys

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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)

##### Definition of column vector, coordinate vector space math.la.d.vec.col.coord

##### Definition of vector, coordinate vector space math.la.d.vec.coord

##### Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

##### math.la.t.vec.sum.geometric.RnCn

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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.)

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### 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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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.

##### Determine if a particular vector is in the span of a set of vectors in R^3 math.la.e.vec.span.of.r3

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn

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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)

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

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In-class activity for linear combinations and span.

##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### Determine if a particular vector is in the span of a set of vectors in R^3 math.la.e.vec.span.of.r3

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

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University of Waterloo Math Online -

##### Definition of vector, coordinate vector space math.la.d.vec.coord

##### Definition of column vector, coordinate vector space math.la.d.vec.col.coord

##### Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

##### Definition of scalar, coordinate vector space math.la.d.scalar

##### Example of a sum of vectors interpreted geometrically in R^2 math.la.e.vec.sum.geometric.r2

##### Example of vector-scalar multiplication in R^2 math.la.e.vec.scalar.mult.r2

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

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Slides for the accompanying video from University of Waterloo.

##### Definition of column vector, coordinate vector space math.la.d.vec.col.coord

##### Definition of vector, coordinate vector space math.la.d.vec.coord

##### Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

##### Definition of scalar, coordinate vector space math.la.d.scalar

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From the University of Waterloo Math Online

##### The vector space properties of R^n (or C^n) math.la.t.vec.axioms.rncn

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn

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Slides from the corresponding video from the University of Waterloo.

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn

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Quiz from the University of Waterloo.

##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

##### Example of vector-scalar multiplication in R^2 math.la.e.vec.scalar.mult.r2

##### Example of linear combination of vectors in R^2 math.la.e.vec.lincomb.r2

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### Determine if a particular vector is in the span of a set of vectors in R^3 math.la.e.vec.span.of.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

##### Determine if a particular set of vectors in R^3 in linearly independent math.la.e.vec.linindep.r3

##### Definition of linear dependence relation math.la.d.vec.lindep.relation

##### A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other. math.la.t.vec.lindep.two

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

##### Definition of linear combination of vectors, arbitrary vector space math.la.d.vec.lincomb.arb

##### Example of linear combination of vectors in R^2 math.la.e.vec.lincomb.r2

##### Example of writing a given vector in R^3 as a linear combination of given vectors math.la.e.vec.lincomb.weight.solve.r3

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##### Definition of column vector, coordinate vector space math.la.d.vec.col.coord

##### Definition of size of a vector, coordinate vector space math.la.d.vec.size.coord

##### Definition of vector, coordinate vector space math.la.d.vec.coord

##### Definition of entry/component of a vector, coordinate vector space math.la.d.vec.component.coord

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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.

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

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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.

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

##### Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord

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##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

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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.

##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

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##### Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord

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##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys

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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.

##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod

##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys

##### 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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##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.row_reduce.i

##### Definition of matrix equation math.la.d.mat.eqn

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For a specific 3x3 matrix, solve Ax=0 by row reducing an augmented matrix.

##### Example of solving a 3-by-3 homogeneous matrix equation math.la.e.mat.eqn.3x3.homog.solve

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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?

##### Definition of matrix multiplication in terms of column vectors math.la.d.mat.mult.col

##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod

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##### Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

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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.

##### Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn

##### Equivalence theorem: the columns of A span R^n (or C^n). math.la.t.equiv.col.span

##### Equivalence theorem: there is a pivot position in every row of A. math.la.t.equiv.row.pivot

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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.

##### Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod

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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.

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##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys

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##### Matrix-scalar product is commutative math.la.t.mat.scalar.prod.commut

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##### Equivalence theorem: the columns of A span R^n (or C^n). math.la.t.equiv.col.span

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##### Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.homog.row_reduce.i

##### Definition of homogeneous linear system of equations math.la.d.linsys.homog

##### A homogeneous system has a nontrivial solution if and only if it has a free variable. math.la.t.linsys.homog.nontrivial

##### Parametric vector form of the solution set of a system of linear equations math.la.c.linsys.soln_set.vec

##### Definition of trivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.trivial

##### Definition of nontrivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.nontrivial

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##### Definition of homogeneous linear system of equations math.la.d.linsys.homog

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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.

##### Definition of homogeneous linear system of equations math.la.d.linsys.homog

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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

##### 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

##### If a set of vectors in R^n (or C^n) contains more than n elements, then the set is linearly dependent. math.la.t.vec.lindep.more.rncn

##### If a set of vectors contains the zero vector, then the set is linearly dependent. math.la.t.vec.lindep.zero

##### 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

##### Definition of linear dependence relation math.la.d.vec.lindep.relation

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Video Lesson from University of Waterloo.

##### Determine if a particular set of vectors in R^3 in linearly independent math.la.e.vec.linindep.r3

##### Definition of linear dependence relation math.la.d.vec.lindep.relation

##### 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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Quiz from the University of Waterloo.

##### Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord

##### Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord

##### Example of vector-scalar multiplication in R^2 math.la.e.vec.scalar.mult.r2

##### Example of linear combination of vectors in R^2 math.la.e.vec.lincomb.r2

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### Determine if a particular vector is in the span of a set of vectors in R^3 math.la.e.vec.span.of.r3

##### Determine if a particular set of vectors in R^3 in linearly independent math.la.e.vec.linindep.r3

##### Definition of linear dependence relation math.la.d.vec.lindep.relation

##### A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other. math.la.t.vec.lindep.two

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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 linear dependence relation math.la.d.vec.lindep.relation

##### math.la.d.vec.lindep.relation.trivial

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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).

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##### Equivalence theorem: the columns of A are linearly independent. math.la.t.equiv.col.linindep

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##### If a set of vectors in R^n (or C^n) contains more than n elements, then the set is linearly dependent. math.la.t.vec.lindep.more.rncn

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Linear transformations are defined, and some small examples (and non examples) are explored. (need tag for R^2 -> R^2 example, general)

##### Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord

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

##### A linear transformation maps zero to zero. math.la.t.lintrans.z

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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Examples of special types of linear transformation from R^2 to R^2: dilation, projection, and shear. (Some issues with the video: things re-start around the 10 second mark, and at 3:46 the word "projection" is said, when it should be "transformation". Also, at the end maybe it could be described why it is called a 'shear'.)

##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2

##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2

##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2

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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.

##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2

##### Matrix-vector multiplication is a linear transformation. math.la.t.mat.vec.mult.lintrans

##### A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord

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

##### Matrices act as a transformations by multiplying vectors math.la.c.transformation.matrix

##### Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord

##### Matrix-vector multiplication is a linear transformation. math.la.t.mat.vec.mult.lintrans

##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2

##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2

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Motivation of the definition of a linear transformation using properties of matrices; examples; geometric intuition; matrix representation of a linear transformation

##### Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord

##### Geometric properties of linear transformations math.la.c.lintrans.geometric

##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2

##### Definition of matrix representation of a linear transformation, coordinate vector space math.la.d.lintrans.mat.repn.coord

##### Non-example of a linear transformation math.la.e.lintrans.not

##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2

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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.

##### A linear transformation maps zero to zero. math.la.t.lintrans.z

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##### A linear transformation of a linear combination is the linear combination of the linear transformation math.la.t.lintrans.lincomb

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##### Matrices act as a transformations by multiplying vectors math.la.c.transformation.matrix

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##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2

##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2

##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2

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Visualize 2-d linear transformations by looking at the image of geometric object. (Need topic: Visualize a linear transformation on R^2 by its effect on a region.)

##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2

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##### Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2

##### Matrix-vector multiplication is a linear transformation. math.la.t.mat.vec.mult.lintrans

##### A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord

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In-class activity to be completed after an introduction to transformations and ideally in teams. In part 1, students are guided to discover the theorem describing the matrix of a linear transformation from R^n to R^m. In part 2, students learn the one-to-one and onto properties of linear transformations, and are asked to relate these properties to the properties of the matrices (linear independence of columns and columns spanning the codomain).

##### Definition of the standard matrix for a linear transformation, coordinate setting math.la.d.lintrans.mat.basis.standard.coord

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##### Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord

##### Geometric properties of linear transformations math.la.c.lintrans.geometric

##### Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2

##### Definition of matrix representation of a linear transformation, coordinate vector space math.la.d.lintrans.mat.repn.coord

##### Non-example of a linear transformation math.la.e.lintrans.not

##### Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2

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##### A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord

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Motivation and definition of the inverse of a matrix

##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of identity matrix math.la.d.mat.identity

##### Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique

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##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult

##### A linear system is equivalent to a vector equation. math.la.t.linsys.vec

##### A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys

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

##### Notation for entry of matrix math.la.d.mat.entry

##### Definition of m by n matrix math.la.d.mat.m_by_n

##### Definition of diagonal matrix math.la.d.mat.diagonal

##### Definition of identity matrix math.la.d.mat.identity

##### Definition of zero matrix math.la.d.mat.zero

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

##### Definition of sum of matrices math.la.d.mat.sum

##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult

##### Matrix addition is commutative and associative. math.la.t.mat.add.commut_assoc

##### Matrix-scalar multiplication is commutative, associative, and distributive. math.la.t.mat.scalar.mult.commut_assoc

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##### Definition of matrix multiplication in terms of column vectors math.la.d.mat.mult.col

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A 2x2 example is used to show that AB does not always equal BA.

##### Matrix multiplication is not commutative in general. math.la.c.mat.mult.commut

##### Example of multiplying 2x2 matrices math.la.e.mat.mult.2x2

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Example 3x2 and a 2x3 matrices are used to show that AB does not always equal BA

##### Matrix multiplication is not commutative in general. math.la.c.mat.mult.commut

##### Example of multiplying nonsquare matrices math.la.e.mat.mult.nonsquare

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

##### Definition of transpose of a matrix math.la.d.mat.transpose

##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose

##### Matrix transpose is an involution. math.la.t.mat.transpose.involution

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

##### Definition of sum of matrices math.la.d.mat.sum

##### Matrix multiplication is not commutative in general. math.la.c.mat.mult.commut

##### For matrices, AB=AC does not imply B=C in general. math.la.c.mat.mult.cancellation

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

##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of identity matrix math.la.d.mat.identity

##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity

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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

##### Matrix-scalar multiplication is commutative, associative, and distributive. math.la.t.mat.scalar.mult.commut_assoc

##### Matrix multiplication is distributive over matrix addition. math.la.t.mat.mult.distributive

##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity

##### Definition of transpose of a matrix math.la.d.mat.transpose

##### Matrix transpose is an involution. math.la.t.mat.transpose.involution

##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose

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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.

##### Definition of identity matrix math.la.d.mat.identity

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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.

##### Definition of transpose of a matrix math.la.d.mat.transpose

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##### Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult

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##### Matrix transpose is an involution. math.la.t.mat.transpose.involution

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##### Definition of sum of matrices math.la.d.mat.sum

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##### Definition of the diagonal of a matrix math.la.d.mat.thediagonal

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##### Definition of equality of matrices math.la.d.mat.equal

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##### Definition of matrix multiplication in terms of column vectors math.la.d.mat.mult.col

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##### Matrix multiplication is associative. math.la.t.mat.mult.assoc

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##### The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose

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##### Matrix multiplication is distributive over matrix addition. math.la.t.mat.mult.distributive

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##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity

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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.

##### Definition of diagonal matrix math.la.d.mat.diagonal

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Motivation and definition of the inverse of a matrix

##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of identity matrix math.la.d.mat.identity

##### Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique

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##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of identity matrix math.la.d.mat.identity

##### The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity

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

##### Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2

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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)

##### 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

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

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Suggested classroom activities on matrix inverses.

##### Definition of matrix inverse math.la.d.mat.inv

##### 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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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.

##### Definition of matrix inverse math.la.d.mat.inv

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

##### Definition of change of corrdinates matrix between two bases, arbitrary vector space math.la.d.vsp.change_of_basis.arb

##### The set of all polynomials of degree at most n is a vector space. math.la.t.vsp.change_of_basis.arb

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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

##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity

##### 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

##### Equivalence theorem: the matrix A has a left inverse. math.la.t.equiv.inv.left

##### Equivalence theorem: the matrix A has a right inverse. math.la.t.equiv.inv.right

##### Matrix inverse is an involution. math.la.t.mat.inv.involution

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

##### Definition of nonsingular matrix: matrix is invertible math.la.d.mat.nonsingular.inv

##### Definition of matrix inverse math.la.d.mat.inv

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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.

##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of nonsingular matrix: matrix is invertible math.la.d.mat.nonsingular.inv

##### Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

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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

##### Matrix inverse is an involution. math.la.t.mat.inv.involution

##### For n-by-n invertible matrices A and B, the product AB is invertible, and (AB)^-1=B^-1 A^-1. math.la.t.mat.inv.shoesandsocks

##### Matrix transpose commutes with matrix inverse. math.la.t.mat.inv.transpose

##### Definition of elementary matrix math.la.d.mat.elementary

##### Row operations are given by multiplication by elementary matrices. math.la.d.mat.mult.elementary

##### Elementary matrices are invertible. math.la.d.mat.elementary.inv

##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity

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##### Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution math.la.d.mat.nonsingular.z

##### Definition of singular matrix math.la.d.mat.singular

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##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity

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

##### Definition of matrix inverse math.la.d.mat.inv

##### math.la.d.mat.invertible

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##### Matrix transpose commutes with matrix inverse. math.la.t.mat.inv.transpose

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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.

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

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##### Matrix inverse is an involution. math.la.t.mat.inv.involution

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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.

##### Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2

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##### For n-by-n invertible matrices A and B, the product AB is invertible, and (AB)^-1=B^-1 A^-1. math.la.t.mat.inv.shoesandsocks

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##### Elementary matrices are invertible. math.la.d.mat.elementary.inv

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##### Definition of elementary matrix math.la.d.mat.elementary

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Motivation and definition of the inverse of a matrix

##### Definition of matrix inverse math.la.d.mat.inv

##### Definition of identity matrix math.la.d.mat.identity

##### Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique

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

##### Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity

##### Equivalence theorem: the matrix A has a left inverse. math.la.t.equiv.inv.left

##### Equivalence theorem: the matrix A has a right inverse. math.la.t.equiv.inv.right

##### Matrix inverse is an involution. math.la.t.mat.inv.involution

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

##### Definition of nonsingular matrix: matrix is invertible math.la.d.mat.nonsingular.inv

##### Definition of matrix inverse math.la.d.mat.inv

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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.

##### Equivalence theorem: the linear transformation given by T(x)=Ax has an inverse. math.la.t.equiv.lintrans.inv

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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

##### Any linearly independent set can be expanded to a basis for the (sub)space, arbitrary vector space. math.la.t.vsp.linindep.basis.arb

##### If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, arbitrary vector space. math.la.t.vsp.dim.linindep.arb

##### If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, arbitrary vector space. math.la.t.vsp.dim.span.arb

##### The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space. math.la.t.vsp.subspace.dim.arb

##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot

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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.

##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot

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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

##### Definition of column space of a matrix math.la.d.mat.col_space

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

##### Definition of kernel of linear transformation, arbitrary vector space math.la.d.lintrans.kernel.arb

##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb

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##### Definition of matrix null space (right) math.la.d.mat.null_space.right

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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 \(\complex{m}\text{.}\)

##### Definition of the standard basis of R^n (or C^n) math.la.d.vsp.basis.standard.rncn

##### Definition of unit vector, coordinate setting math.la.d.vec.unit.coord

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##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot

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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.

##### The null space of a matrix is a subspace of R^n (or C^n). math.la.t.mat.null_space.rncn

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##### Definition of column space of a matrix math.la.d.mat.col_space

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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.

##### If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity

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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

##### Each vector can be written uniquely as a linear combination of vectors from a given basis. math.la.t.vsp.basis.coord.unique

##### Definition of coordinates relative to a given basis, arbitrary vector space math.la.d.vsp.basis.relative.arb

##### Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n) math.la.d.vsp.basis.coord.change.rncn

##### Definition of coordinates relative to a given basis, coordinate vector space math.la.d.vsp.basis.relative.coord

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##### Equivalence theorem: the null space of the matrix A is {0}. math.la.t.equiv.nullspace

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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.

##### Equivalence theorem: the matrix A has rank n. math.la.t.equiv.rank

##### Equivalence theorem: the nullity of the matrix A is zero. math.la.t.equiv.nullity

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##### Definition of rank of a matrix math.la.d.mat.rank

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##### If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity

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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.

##### Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations

##### Equivalence theorem: the determinant of A is nonzero. math.la.t.equiv.det

##### The determinant of a triangular matrix is the product of the entries on the diagonal. math.la.t.mat.det.trianglar

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##### Definition of determinant of a matrix as a cofactor expansion across the first row math.la.d.mat.det.cofactor

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##### Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations

##### Equivalence theorem: the determinant of A is nonzero. math.la.t.equiv.det

##### The determinant of a triangular matrix is the product of the entries on the diagonal. math.la.t.mat.det.trianglar

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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

##### A matrix and its transpose have the same determinant. math.la.t.mat.det.transpose

##### If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product

##### Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations

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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.

##### Equivalence theorem: the determinant of A is nonzero. math.la.t.equiv.det

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##### Theorem describing the determinants of elementary matrices. math.la.t.mat.elementary.det

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##### If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product

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##### A matrix and its transpose have the same determinant. math.la.t.mat.det.transpose

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

##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb

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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...

##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb

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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.

##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb

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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.

##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb

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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.

##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb

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

##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb

##### Definition of vector, arbitrary vector space math.la.d.vec.arb

##### Definition of vector addition, arbitrary vector space math.la.d.vec.add.arb

##### Definition of vector-scalar multiplication, arbitrary vector space math.la.d.vec.scalar.mult.arb

##### The additive inverse of a vector is called the negative of the vector. math.la.d.vsp.vector.negative

##### The zero scalar multiplied by any vector equals the zero vector. math.la.t.vsp.scalar.mult.z

##### The zero vector multiplied by any scalar equals the zero vector. math.la.t.vsp.vector.mult.z

##### The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative

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##### Definition of linear combination of vectors, arbitrary vector space math.la.d.vec.lincomb.arb

##### Example of linear combination of vectors in R^2 math.la.e.vec.lincomb.r2

##### Example of writing a given vector in R^3 as a linear combination of given vectors math.la.e.vec.lincomb.weight.solve.r3

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

##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb

##### Definition of subspace spaned by a set of vectors, arbitrary vector space math.la.d.vec.span.subspace.arb

##### Definition of zero subspace, arbitrary vector space math.la.d.vsp.subspace.z

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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.

##### The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative

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##### math.la.e.vsp.polynomial.leq_n

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##### The zero scalar multiplied by any vector equals the zero vector. math.la.t.vsp.scalar.mult.z

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##### Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb

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##### Definition of vector addition, arbitrary vector space math.la.d.vec.add.arb

##### Definition of vector-scalar multiplication, arbitrary vector space math.la.d.vec.scalar.mult.arb

##### Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb

##### Definition of vector, arbitrary vector space math.la.d.vec.arb

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##### math.la.e.vsp.function

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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.

##### Definition of spanning set for a subspace, arbitrary vector space math.la.d.vsp.span.set.arb

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##### The zero vector multiplied by any scalar equals the zero vector. math.la.t.vsp.vector.mult.z

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##### Definition of span of a set of vectors, arbitrary vector space math.la.d.vec.span.arb

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##### Definition of linear combination of vectors, arbitrary vector space math.la.d.vec.lincomb.arb

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

##### A linear transformation maps zero to zero. math.la.t.lintrans.z

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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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...

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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##### Definition of matrix inverse math.la.d.mat.inv

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

##### Definition of change of corrdinates matrix between two bases, arbitrary vector space math.la.d.vsp.change_of_basis.arb

##### The set of all polynomials of degree at most n is a vector space. math.la.t.vsp.change_of_basis.arb

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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##### Definition of column space of a matrix math.la.d.mat.col_space

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

##### Definition of kernel of linear transformation, arbitrary vector space math.la.d.lintrans.kernel.arb

##### Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb

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##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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##### 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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##### 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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##### math.la.d.vec.lindep.relation.rep

##### math.la.d.vec.lindep.relation.trvial.rep

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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.

##### Definition of the standard basis of the polynomials of degree at most n math.la.d.vsp.basis.standard.leq_n

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##### Definition of basis of a vector space (or subspace), arbitrary vector space math.la.d.vsp.basis.arb

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##### Each vector can be written uniquely as a linear combination of vectors from a given basis. math.la.t.vsp.basis.coord.unique

##### Definition of coordinates relative to a given basis, arbitrary vector space math.la.d.vsp.basis.relative.arb

##### Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n) math.la.d.vsp.basis.coord.change.rncn

##### Definition of coordinates relative to a given basis, coordinate vector space math.la.d.vsp.basis.relative.coord

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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.

##### Definition of isomorphism between vector spaces math.la.d.vsp.isomorphism

##### The coordinate vector/mapping relative to a given basis is a bijective linear mapping to R^n (or C^n). math.la.t.vsp.basis.coord.vector.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.

##### Definition of coordinate vector/mapping relative to a given basis, arbitrary vector space math.la.d.vsp.basis.coord.vector.arb

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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.

##### Definition of isomorphism between vector spaces math.la.d.vsp.isomorphism

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##### Each vector can be written uniquely as a linear combination of vectors from a given basis. math.la.t.vsp.basis.coord.unique

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##### Any linearly independent set can be expanded to a basis for the (sub)space, arbitrary vector space. math.la.t.vsp.linindep.basis.arb

##### If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, arbitrary vector space. math.la.t.vsp.dim.linindep.arb

##### If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, arbitrary vector space. math.la.t.vsp.dim.span.arb

##### The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space. math.la.t.vsp.subspace.dim.arb

##### The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot

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##### Definition of dimension of a vector space (or subspace), arbitrary vector space math.la.d.vsp.dim.arb

##### Definition of dimension of a vector space (or subspace) being finite or infinite, arbitrary vector space math.la.d.vsp.dim.finite_infinite.arb

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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 \(\complex{m}\text{.}\)

##### The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space. math.la.t.vsp.subspace.dim.arb

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##### 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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##### 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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##### 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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##### Definition of nullity of a matrix math.la.d.mat.nullity

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##### Row equivalent matrices have the same row space. math.la.d.mat.row_space.row_equiv

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##### The row space and the column space of a matrix have the same dimension. math.la.t.mat.row_space.col_space

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##### Definition of row space of a matrix math.la.d.mat.row_space

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##### Definition of matrix inverse math.la.d.mat.inv

##### The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv

##### Definition of change of corrdinates matrix between two bases, arbitrary vector space math.la.d.vsp.change_of_basis.arb

##### The set of all polynomials of degree at most n is a vector space. math.la.t.vsp.change_of_basis.arb

##### Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb

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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.

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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.

##### Definition of eigenvector(s) of a matrix math.la.d.mat.eigvec

##### Definition of eigenvalue(s) of a matrix math.la.d.mat.eig

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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

##### Eigenvectors with distinct eigenvalues are linearly independent. math.la.t.mat.eigvec.linindep

##### Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp

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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.

##### Definition of characteristic equation of a matrix math.la.d.mat.charpoly.eqn

##### Definition of characteristic polynomial of a matrix math.la.d.mat.charpoly

##### The eigenvalues of a triangular matrix are the entries on the main diagonal. math.la.t.mat.eig.triangular

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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.

##### Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp

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##### Definition of eigenvector(s) of a matrix math.la.d.mat.eigvec

##### Definition of eigenvalue(s) of a matrix math.la.d.mat.eig

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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.

##### Eigenvectors with distinct eigenvalues are linearly independent. math.la.t.mat.eigvec.linindep

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##### Definition of characteristic equation of a matrix math.la.d.mat.charpoly.eqn

##### Definition of characteristic polynomial of a matrix math.la.d.mat.charpoly

##### The eigenvalues of a triangular matrix are the entries on the main diagonal. math.la.t.mat.eig.triangular

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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.

##### Definition of similar matrices math.la.d.mat.similar

##### Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig

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##### Definition of characteristic polynomial of a matrix math.la.d.mat.charpoly

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##### Equivalence theorem: the matrix A does not have zero as an eigenvalue. math.la.t.equiv.eig

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##### Definition of similar matrices math.la.d.mat.similar

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##### Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig

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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.

##### Definition of diagonalizable matrix math.la.d.mat.diagonalizable

##### An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors. math.la.t.mat.diagonalizable

##### A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns. math.la.t.mat.diagonalized_by

##### An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct

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##### An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors. math.la.t.mat.diagonalizable

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##### The dimension of a eigenspace is less than or equal to the multiplicity of the eigenvalue. math.la.t.mat.eig.multiplicity.eigenspace

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##### An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct

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##### 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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##### Definition of diagonalizable matrix math.la.d.mat.diagonalizable

##### Definition of matrix diagonalization math.la.d.mat.diagonalization

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##### Definition of matrix representation of a linear transformation with respect to bases of the spaces, arbitrary vector space math.la.d.lintrans.mat.repn.arb

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- 3
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##### A matrix with real entries has eigenvalues occurring in conjugate pairs. math.la.t.mat.real.eig.cn

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- GFDL-1.2
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- September 11th, 2017
- 3 years ago
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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.

##### Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.coord

##### Definition of inner product, real entries, coordinate setting math.la.d.innerproduct.real.coord

##### The standard inner product is commutative, coordinate setting. math.la.t.innerproduct.commutative.coord

##### The standard inner product distributes over addition, coordinate setting. math.la.t.innerproduct.distributive.coord

##### The standard inner product commutes with real scalar multiplication, coordinate setting. math.la.t.innerproduct.commutative.scalar.real.coord

##### The standard inner product of a vector with itself is zero only for the zero vector, coordinate setting. math.la.t.innerproduct.self.z.coord

##### Definition of a vector being orthogonal to a subspace math.la.d.vec.subspace.orthogonal

##### Definition of two vectors being orthogonal math.la.d.vec.orthogonal

##### Definition of unit vector, coordinate setting math.la.d.vec.unit.coord

##### Definition of cross product math.la.d.crossproduct

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- October 23rd, 2013
- 7 years ago
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- 3
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- Video
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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.

##### Definition of unit vector, coordinate setting math.la.d.vec.unit.coord

##### Definition of distance, coordinate setting math.la.d.distance.coord

##### Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.coord

##### The standard inner product of a vector with itself is zero only for the zero vector, coordinate setting. math.la.t.innerproduct.self.z.coord

##### The standard inner product commutes with real scalar multiplication, coordinate setting. math.la.t.innerproduct.commutative.scalar.real.coord

##### The standard inner product distributes over addition, coordinate setting. math.la.t.innerproduct.distributive.coord

##### The standard inner product is commutative, coordinate setting. math.la.t.innerproduct.commutative.coord

##### Definition of inner product, real entries, coordinate setting math.la.d.innerproduct.real.coord

##### Definition of two vectors being orthogonal math.la.d.vec.orthogonal

- Created On
- October 23rd, 2013
- 7 years ago
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- 3
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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.

##### Definition of inner product, real entries, coordinate setting math.la.d.innerproduct.real.coord

##### The standard inner product is commutative, coordinate setting. math.la.t.innerproduct.commutative.coord

##### The standard inner product distributes over addition, coordinate setting. math.la.t.innerproduct.distributive.coord

##### The standard inner product of a vector with itself is zero only for the zero vector, coordinate setting. math.la.t.innerproduct.self.z.coord

##### Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.coord

##### Definition of unit vector, coordinate setting math.la.d.vec.unit.coord

##### Definition of distance, coordinate setting math.la.d.distance.coord

##### Definition of two vectors being orthogonal math.la.d.vec.orthogonal

##### Two vectors are orthogonal if and only if the Pythagorean Theorem holds. math.la.t.vec.orthogonal

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

##### Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.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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- English
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- text/html

##### The standard inner product of a vector with itself is non-negative, coordinate setting. math.la.t.innerproduct.self.positive.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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- English
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- text/html

##### Definition of the standard basis of R^n (or C^n) math.la.d.vsp.basis.standard.rncn

##### Definition of unit vector, coordinate setting math.la.d.vec.unit.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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- Language
- English
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- text/html

This is from the University of Waterloo. It includes content about Projections, as well as some content from the Multivariable Calculus. These notions are developed in Euclidean Space.

##### Formula for the (orthogonal) projection of one vector onto another vector math.la.t.vec.projection

##### Definition of (orthogonal) projection of one vector onto another vector math.la.d.vec.projection

- Created On
- October 23rd, 2013
- 7 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

This is a quiz from the University of Waterloo. It is a quiz about projections that is strictly in R^n. It additionally asks questions on perpendicular vectors and cross products.

##### Definition of (orthogonal) projection of one vector onto another vector math.la.d.vec.projection

##### Formula for the (orthogonal) projection of one vector onto another vector math.la.t.vec.projection

- Created On
- October 23rd, 2013
- 7 years ago
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- 2
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- Unknown
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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

##### Definition of orthonormal set of vectors math.la.d.vec.orthonormal_set

##### A matrix with real entries and orthonormal columns preserves norms. math.la.t.mat.col.orthonormal.norm.rn

##### A matrix with real entries and orthonormal columns preserves dot products. math.la.t.mat.col.orthonormal.dot.rn

##### Definition of orthogonal matrix math.la.d.mat.orthogonal

##### Definition of orthonormal basis of a (sub)space math.la.d.subspace.basis.orthonormal

##### A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose. math.la.t.mat.col.orthonormal.inv.rn

- 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

##### An orthogonal set of nonzero vectors is linearly independent. math.la.t.vec.orthogonal_set.linindep

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

##### Definition of orthonormal set of vectors math.la.d.vec.orthonormal_set

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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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- text/html

##### Definition of orthogonal set of vectors math.la.d.vec.orthogonal_set

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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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- English
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- text/html

##### Formula for the coordinates of a vector with respect to an orthogonal basis. math.la.t.subspace.basis.orthogonal

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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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Orthogonal projection onto subspace in R^n minimizes distance; projection formula simplification for orthonormal bases; relation to orthogonal matrices

##### The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector. math.la.t.vec.projection.closest

##### Formula for the coordinates of the projection of a vector onto a subspace, with respect to an orthonormal basis. math.la.t.subspace.basis.orthonormal

##### Definition of orthogonal projection onto a subspace math.la.d.vec.projection.subspace

- Created On
- August 21st, 2017
- 3 years ago
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- 4
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- Video
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- English
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- text/html; charset=utf-8

##### Description of the Gram-Schmidt process math.la.d.gramschmidt

##### The Gram-Schmidt process converts a basis into an orthogonal basis. math.la.t.gramschmidt

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- GFDL-1.2
- Submitted At
- September 11th, 2017
- 3 years ago
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- 3
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- English
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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.

##### A matrix is orthogonally diagonalizable if and only if it is symmetric. math.la.t.mat.diagonalizable.orthogonally

##### Definition of matrix diagonalization math.la.d.mat.diagonalization

##### Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal. math.la.t.mat.symmetric.eig.orthogonal

##### The spectral theorem for symmetric matrices math.la.t.mat.symmetric.spectral

- 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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##### Definition of symmetric matrix math.la.d.mat.symmetric

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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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- English
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- text/html