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quiz2 - Quiz#2 1 Let V and W be ﬁnite-dimensional vector...

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Quiz #2 1. Let V and W be finite-dimensional vector spaces (over F ) with ordered bases α and β , respectively. And let T be a linear transformation from V to W . Label the following statements as being true or false. No verification is needed. (a) Given x 1 , x 2 V and y 1 , y 2 W , there exists a linear transfor- mation U : V W such that U ( x 1 ) = y 1 and U ( x 2 ) = y 2 . (Solution) False (b) [ T ( v )] β = [ T ] β α [ v ] α for all v V . (Solution) True (c) T is invertible if and only if T is one-to-one and onto. (Solution) True (d) Let A and B are matrices. Then AB = I implies that A and B are invertible. (Solution) False (e) Let U be a linear operator on V . Then for any ordered bases γ and γ for V , [ U ] γ is similar to [ U ] γ . (Solution) True 2. Let T : R 2 R 3 be defined by T ( a 1 , a 2 ) = ( a 1 - a 2 , a 1 , 2 a 1 + a 2 ). Let α = { (1 , 2) , (2 , 3) } and γ = { (1 , 1 , 0) , (0 , 1 , 1) , (2 , 2 , 3) } . Compute [ T ] γ α . (Solution) T (1 , 2) = ( - 1 , 1 , 4) = - 7 3 (1 , 1 , 0) + 2(0 , 1 , 1) + 2 3 (2 , 2 , 3) , T (2 , 3) = ( - 1 , 2 , 7) = - 11 3 (1 , 1 , 0) + 3(0 , 1 , 1) + 4 3 (2 , 2 , 3) . Therefore [ T ] γ α = - 7 3 - 11 3 2 3 2 3 4 3 . 1

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3. Find the change of coordinate matrix that changes
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