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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....
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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.
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