lecture_5 - MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 16...

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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 16 Matrix Multiplication Matrix-Vector Product Expressed as Columns. Say that we have an m n matrix and a n 1 matrix: A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn and c = c 1 c 2 . . . c n . We will show that the product of these two can be expressed as a linear combination of the columns of A : A c = n X j =1 c j col j ( A ) where col j ( A ) = a 1 j a 2 j . . . a mj . To see why, we recall the definition of matrix multiplication: A c = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn c 1 c 2 . . . c n = a 11 c 1 + a 12 c 2 + + a 1 n c n a 21 c 1 + a 22 c 2 + + a 2 n c n . . . a m 1 c 1 + a m 2 c 2 + + a mn c n = a 11 c 1 a 21 c 1 . . . a m 1 c 1 + a 12 c 2 a 22 c 2 . . . a m 2 c 2 + + a 1 n c n a 2 n c n . . . a mn c n = c 1 a 11 a 21 . . . a m 1 + c 2 a 12 a 22 . . . a m 2 + + c n a 1 n a 2 n . . . a mn = c 1 col 1 ( A ) + c 2 col 2 ( A ) + + c n col n ( A ) . Example. We give an example of this. Consider the following matrices: A = 2- 1- 3 4 2- 2 and c = 2- 3 4 . Using the definition of the matrix product, we have A c = (2)(2) + (- 1)(- 3) + (- 3)(4) (4)(2) + (2)(- 3) + (- 2)(4) =- 5- 6 ....
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lecture_5 - MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 16...

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