10w_math115a_hw6_solutions

10w_math115a_hw6_solutions - Math 115A Homework # 6...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 115A Homework # 6 Solutions Prof. Yehuda Shalom TA: Darren Creutz Date Due: 25 February 2010 Problem 1. Let V = P 2 ( R ) and = { 1 , x, x 2 } and = { x 2 + 2 x, x + 1 , x- 1 } , both of which are bases for V . (i) Compute the change of coordinate matrix Q from into (find Q such that [ v ] = Q [ v ] for each v V ). Let T : V V be defined by T ( f ( x )) = ( x + 1) f ( x ) where f is the derivative. (ii) Prove that T is a linear transformation. (iii) Compute directly the matrix representation [ T ] . (iv) Compute directly the matrix representation [ T ] . (v) Verify that [ T ] = Q- 1 [ T ] Q . Let U 1 : P 2 ( R ) P 1 ( R ) be defined by U 1 ( f ( x )) = f ( x ) and U 2 : P 1 ( R ) P 2 ( R ) be defined by U 2 ( f ( x )) = ( x + 1) f ( x ) Let 1 be the standard basis for P 1 ( R ) and 2 the standard basis for P 2 ( R ) (so 2 = above). (vi) Compute directly the matrix representations [ U 1 ] 1 2 and [ U 2 ] 2 1 . (vii) Verify that [ T ] 2 2 = [ U 2 ] 2 1 [ U 1 ] 1 2 . Proof. (i) We compute Q = [ Id ] . Since is the standard basis this just means writing in matrix form: Q = 0 1- 1 2 1 1 1 0 1 by putting across the top and writing each in terms of along the side....
View Full Document

This note was uploaded on 03/15/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Winter '10 term at UCLA.

Page1 / 5

10w_math115a_hw6_solutions - Math 115A Homework # 6...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online