assign6 - N b s t L = sL tL 4 Let A be an m × n matrix...

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Math 136 Assignment 6 Due: Wednesday, Mar 2nd 1. Find a spanning set for the kernel and range for each of the following linear mappings. a) f ( x 1 ,x 2 ,x 3 ) = ( x 1 ,x 2 ,x 1 + x 2 - x 3 ) b) f ( x 1 ,x 2 ,x 3 ) = (0 ,x 1 - x 2 ) c) f ( x 1 ,x 2 ,x 3 ,x 4 ) = (0 , 0) d) f ( x 1 ,x 2 ) = ( x 1 + 2 x 2 , 2 x 1 + 4 x 2 , 3 x 1 + 6 x 2 ) 2. Let L : R n R m be a linear mapping. Prove that L ( ~ 0) = ~ 0. 3. Let L , M , and N be linear mappings from R n to R m and let s,t R be scalars. Prove that a) L + ( M + N ) = ( L + M ) +
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Unformatted text preview: N . b) ( s + t ) L = sL + tL . 4. Let A be an m × n matrix. Prove that for any ~a ∈ Col( A ) and ~x ∈ Null( A T ) we have ~a · ~x = 0. 5. Let L : R n → R m and M : R m → R p be linear mappings. Prove that M ◦ L is a linear mapping and [ M ◦ L ] = [ M ][ L ] 1...
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This note was uploaded on 07/16/2011 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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