115hw2solutions - Math 115A Homework 2 Due October 24th...

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Math 115A Homework 2 Due October 24th, 2008 1. For each of the following maps T : V W , check if T is an F -linear transformation. If yes, write out a proof. If not, prove that. (2 pts each) a) F = R , V = R 3 , W = R 2 and T ( a 1 ,a 2 ,a 3 ) = ( a 1 a 2 ,a 3 + 2 a 1 ). Let λ = 2 R and a = ( a 1 ,a 2 ,a 3 ) = (1 , 1 , 1). Then T ( λ a ) = T (2 , 2 , 2) = (4 , 6) but λT ( a ) = (2 , 6). That is, T does not commute with scalar multiplication and is therefore not a linear transformation. b) F = C , V = C 2 , W = C and T ( z,w ) = z . Let λ = i C and v = (1 , 0) V . Then T ( λ v ) = - i but λT ( v ) = i . That is, T does not commute with scalar multiplication and is therefore not a linear transformation. c) F = R , V = C ( R ) the set of continuous real-valued functions on the real numbers, W = R and T ( f ) = R 1 0 x 2 f ( x ) dx . The map U : V V that is defined by U ( f )( x ) = x 2 f ( x ) is a linear transformation because multiplication of real numbers is commutative, and multiplication and addition are distributive. Moreover, the map S : V W = R defined by S ( f ) = R 1 0 f ( x ) dx
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115hw2solutions - Math 115A Homework 2 Due October 24th...

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