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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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- GFDL-1.2
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- September 11th, 2017
- 6 years ago
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- English
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- License
- GFDL-1.2
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- September 11th, 2017
- 6 years ago
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- English
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- text/html

- License
- GFDL-1.2
- Submitted At
- September 11th, 2017
- 6 years ago
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- 4
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- English
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- text/html

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

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

- License
- GFDL-1.2
- Submitted At
- September 11th, 2017
- 6 years ago
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- 3
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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

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- August 25th, 2017
- 6 years ago
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- 3
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- Video
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- English
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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

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- August 25th, 2017
- 6 years ago
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- 2
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- Video
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- English
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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?

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- February 17th, 2017
- 6 years ago
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- 2
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- Video
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- Review
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- English
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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.

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

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- October 23rd, 2013
- 9 years ago
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- 3
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Definition of sum of matrices, product of a scalar and a matrix

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- February 17th, 2017
- 6 years ago
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- 3
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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

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- August 25th, 2017
- 6 years ago
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- 3
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- Video
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- English
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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.

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- October 23rd, 2013
- 9 years ago
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- 3
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- Video
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- English
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- GFDL-1.2
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- September 11th, 2017
- 6 years ago
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- 3
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- English
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A rotatable model of the cross product of two vectors.

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- CC-BY-SA-4.0
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- August 7th, 2018
- 5 years ago
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- 2
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- Applet
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- English
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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.

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- August 22nd, 2017
- 6 years ago
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- 2
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- Video
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- English
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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

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- August 25th, 2017
- 6 years ago
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- 3
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- Video
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- English
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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.

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- CC-BY-SA-4.0
- Created On
- February 15th, 2017
- 6 years ago
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- 3
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- Video
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- English
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