lect136_5_w09

lect136_5_w09 - Wednesday January 14 Lecture 5 Matrix...

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Wednesday January 14 Lecture 5 : Matrix algebra II. (Refers to section 3.1) Expectations: 1. Define matrix multiplication 2. Perform block multiplication. 5.0 Example If A is a 3 × 3 matrix with rows vectors r A 1 , r A 2 , r A 3 n . and B is an 3 × 3 matrix with column vectors c B 1 , c B 2 , c B 3 . Find an expression which represents the ( i , j ) th entry of B T A T in terms of the rows and columns of A and B . If the entries of A are 1, 2, 3, …., 9 (listed horizontally) and the entries of B are 10, 11, 12,… 18, find the (2, 3)th entry of B T A T . Solution: Known: the rows of B T are the columns of B . So B T has rows c B 1 , c B 2 , c B 3 . Known: the columns of A T are the rows of A . So A T has columns r A 1 , r A 2 , r A 3 . Let D = [ d ij ] = B T A T . Then by definition of the product of matrices d ij = <row i of B T , column j of A T > . = < c Bi , r Aj .> . We seek the entry d 23 . We have d 23 = < c B 2 , r A 3 .> = < (11 , 14, 17) , (7 , 8, 9)> = 77 + 112 + 153 = 342. 5.1 Remark We now show that the dot-product of two vectors is a number that can be expressed as a product of two matrices. o
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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lect136_5_w09 - Wednesday January 14 Lecture 5 Matrix...

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