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chapter3.page04

# chapter3.page04 - • 3 ‚ a The following fact gives...

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4 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES in 3-dimension: Let x = £ x 1 x 2 x 3 / and y = y 1 y 2 y 3 , the dot product of x and y is, x · y = x 1 y 1 + x 2 y 2 + x 3 y 3 Definition 1.3 . Matrix product Let A = ( a ij ) and B = ( b ij ) , if the number of columns of A is the same as number of rows of B , then the product of A and B is given by AB = ( c ij ) where c ij is dot product of i th row of A with j th column of B. Example 1.2 . Let A = 2 3 - 1 4 and B = 0 5 3 - 4 , find AB Solution AB = 2 3 - 1 4 ‚ • 0 5 3 - 4 = 2 · (0) + 3 · (3) 2 · 5 + 3 · ( - 4) ( - 1) · (0) + 4 · 3 - 1 · 5 + 4 · ( - 4) = 9 - 2 12 - 21 Notice, the first element of AB is 2 · (0) + 3 · (3) which is the dot product of first row of A, £ 2 3 / and first column of B,
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Unformatted text preview: • 3 ‚ a The following fact gives properties of matrix product, Theorem 1.2 . Let A,B,C be three matrices and r be a scalar, we have • A ( BC ) = ( AB ) C, r ( AB ) = A ( rB ) (associativity) • A ( B + C ) = AB + AC (distributivity) Notice, in general AB 6 = BA , that is for most of the times, AB is not equal to BA. Using the matrix notation and matrix product, we can write the following system of equations ‰ ax 1 + bx 2 = y 1 cx 1 + dx 2 = y 2...
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