assign6_practice

# assign6_practice - ³± 1-1 ²´ and whose range is Span 3...

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Math 136 Practice Problems # 6 1. Suppose that S and T are linear mappings with matrices [ S ] = 4 - 3 1 1 5 - 3 - 2 0 [ T ] = ± 4 0 2 3 - 2 1 3 0 ² . a) Determine the domain and codomain of each mapping. b) Determine the standard matrices that represent S T and T S . 2. Let L : R n R m be a linear mapping and let s,t R be scalars. Prove that s ( tL ) = ( st ) L . 3. Find a spanning set for the kernel and range of the following linear mappings. a) f ( x 1 ,x 2 ,x 3 ) = ( x 1 - x 2 ,x 1 - 2 x 3 ) b) f ( x 1 ,x 2 ) = ( x 2 , 0 ,x 1 - x 2 ,x 1 ). c) f ( x 1 ,x 2 ,x 3 ) = ( x 3 ,x 1 + x 2 ). 4. Let A be an n × n matrix such that A 2 = 0. Prove that the columnspace of A is a subset of the nullspace of A . 5. Determine a matrix of a linear mapping L : R 2 R 3 whose kernel is Span
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Unformatted text preview: ³± 1-1 ²´ and whose range is Span 3 5 . 6. Let { ~v 1 ,...,~v k } be a set in R n such that Span { ~v 1 ,...,~v k } = R n . Let L : R n → R m be a linear mapping such that Range( L ) = R m . Prove that Span { L ( ~v 1 ) ,...,L ( ~v k ) } = R m . 7. Let T : R n → R m be a linear mapping. Suppose that { ~v 1 ,...,~v k } a linearly dependent set in R n . Prove that { T ( ~v 1 ) ,...,T ( ~v k ) } is a linearly dependent set in R m . 1...
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## This note was uploaded on 07/13/2011 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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