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Column and Row Spaces - A First Course in Linear Algebra

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

math.la.t.mat.eqn.lincomb
GFDL-1.2
Submitted At
September 11th, 2017
7 years ago
Views
2
Type
Textbook
Language
English
Content Type
text/html

Column and Row Spaces - A First Course in Linear Algebra

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

math.la.d.mat.row_space.row_equiv
GFDL-1.2
Submitted At
September 11th, 2017
7 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Column and Row Spaces - A First Course in Linear Algebra

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

math.la.t.mat.col_space.pivot
GFDL-1.2
Submitted At
September 11th, 2017
7 years ago
Views
4
Type
Textbook
Language
English
Content Type
text/html

Column and Row Spaces - A First Course in Linear Algebra

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

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

Column and Row Spaces - A First Course in Linear Algebra

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

math.la.t.mat.row_space.pivot
GFDL-1.2
Submitted At
September 11th, 2017
7 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

Column and Row Spaces - A First Course in Linear Algebra

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

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

Matrix inverses

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

math.la.t.equiv.identity
Created On
August 25th, 2017
7 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Matrix operations

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

math.la.t.mat.mult.transpose
Created On
August 25th, 2017
7 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

How to multiply matrices

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?

math.la.d.mat.vec.prod
Created On
February 17th, 2017
8 years ago
Views
2
Type
Video
Timeframe
Review
Language
English
Content Type
text/html; charset=utf-8

Projections Quiz

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.

math.la.t.vec.projection
Created On
October 23rd, 2013
11 years ago
Views
2
Type
Unknown
Timeframe
Post-class
Perspective
Example
Language
English
Content Type
text/html;charset=UTF-8

Dot Product, Cross Product, and Scalar Equations Quiz

Quiz from the University of Waterloo. This is intended to be used after the video of the same name.

math.la.d.vec.orthogonal
Created On
October 23rd, 2013
11 years ago
Views
3
Type
Unknown
Timeframe
Post-class
Perspective
Example
Language
English
Content Type
text/html;charset=UTF-8

Matrix Operations: Sums Scalar Multiplication

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

math.la.t.mat.scalar.mult.commut_assoc
Created On
February 17th, 2017
8 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8

Orthonormality

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

math.la.t.mat.col.orthonormal.inv.rn
Created On
August 25th, 2017
7 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Dot Product and Cross Product Lesson

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.

math.la.d.crossproduct
Created On
October 23rd, 2013
11 years ago
Views
3
Type
Video
Perspective
Introduction
Language
English
Content Type
text/html;charset=UTF-8

Column and Row Spaces - A First Course in Linear Algebra

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

math.la.t.equiv.col.span
GFDL-1.2
Submitted At
September 11th, 2017
7 years ago
Views
3
Type
Textbook
Language
English
Content Type
text/html

The cross product - Math Insight

A rotatable model of the cross product of two vectors.

math.la.d.crossproduct
CC-BY-SA-4.0
Created On
August 7th, 2018
6 years ago
Views
2
Type
Applet
Language
English
Content Type
text/html; charset=utf-8

Inner products and distance

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.

math.la.t.vec.orthogonal
Created On
August 22nd, 2017
7 years ago
Views
2
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Determinants and their relation to column operations and products

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

math.la.t.mat.det.elementaryoperations
Created On
August 25th, 2017
7 years ago
Views
3
Type
Video
Language
English
Content Type
text/html; charset=utf-8

Matrix Equation: Matrix-Vector Product

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

math.la.t.mat.eqn.linsys
CC-BY-SA-4.0
Created On
February 15th, 2017
8 years ago
Views
3
Type
Video
Timeframe
Pre-class
Perspective
Introduction
Language
English
Content Type
text/html; charset=utf-8