Chapter 1.2 MatrixAlgebra

# Proof write a as column vectors and use properties of

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Unformatted text preview: Proof. Write A as column vectors and use properties of matrix addition and scalar multiplication. Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Matrix Multiplication Let A be an m × n matrix, and let B be a n × k matrix. Let v1 , v2 , . . . , vk be the column vectors of B . For each 1 ≤ i ≤ k , set wi = Avi , a column vector of size n. The product of A and B is the m × k matrix AB deﬁned by AB = [ w1 w2 · · · wk ] = [ Av1 Av2 · · · Avk ]. Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Examples Example Given A= ﬁnd AB . 2 0 −1 3 1 −5 1 2 −1 3 0 2 −2 and B = −1 2 −2 3 0 Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Properties of Matrix Multiplication Theorem If A = [aij ] is an m × n matrix with row vectors w1 , w2 , . . . , wn , and B = [bij ] is an n × k matrix with column vectors v1 , v2 , . . . , vk , then AB = [cij ], where m cij =...
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