assign6_practice

assign6_practice - 1-1 and whose range is Span 3 5 . 6. Let...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1-1 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...
View Full Document

Ask a homework question - tutors are online