A3 - . 3. Let L be a linear operator on an n dimensional...

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

View Full Document Right Arrow Icon
Math 235 Assignment 3 Due: Wednesday, May 26th 1. For each of the following pairs of vector spaces, define an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) P 3 and R 4 . b) The vector space P = { p ( x ) P 3 | p (1) = 0 } and the vector space U of 2 × 2 upper triangular matrices. 2. Suppose that { ~v 1 ,...,~v r } is a linearly independent set in a vector space V and that L : V W is a one-to-one linear map. Prove that { L ( ~v 1 ) ,...,L ( ~v r ) } is a linearly independent set in W
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . 3. Let L be a linear operator on an n dimensional vector space V . Prove that the following are equivalent. 1) L-1 exists. 2) L is one-to-one. 3) null( L ) = { ~ } 4) L is onto. 4. Let V be an n dimensional vector space with basis B and let S be the vector space of all linear operators L : V V . Dene T : S M ( n,n ) by T ( L ) = [ L ] B . a) Prove that T is an isomorphism. b) Use a) to nd a basis for S ....
View Full Document

This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

Ask a homework question - tutors are online