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midtermsolutions - Midterm Solutions Josh Hernandez...

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Midterm Solutions Josh Hernandez November 4, 2009 1 (20 points) Compute the matrix representation [ T ] γ β of the linear transformation T : R 2 R 2 with mapping T ( x, y ) = ( x - y, 2 x + y ) with respect to the ordered bases β = { (1 , 0) , (0 , 1) } and γ = { (-1 , 1) , (1 , 0) } . Solution: First compute T ( β ): T (1 , 0) = (1 - 0 , 2(1) + 0) = (1 , 2) and T (0 , 1) = (0 - 1 , 2(0) + 1) = (-1 , 1) . Now, we write each of those vectors as linear combinations in γ : (1 , 2) = 2(-1 , 1) + 3(1 , 0) and (-1 , 1) = 1(-1 , 1) + 0(1 , 0) . We write these coefficients vertically as the columns of our matrix: [ T ] γ β = 2 1 3 0 . 2 (20 points) Define W = { f P 3 ( R ) | f 0 (0) = 0 } a Prove that W is a subspace of P 3 ( R ). Solution: Take f, g W and k R . By linearity of derivatives, ( f + kg ) 0 (0) = f 0 (0) + kg 0 (0) = 0 + k 0 = 0 . Thus f + kg W , and we have proved the subspace criterion. b Find a basis for W (and prove that it is a basis). Solution: To find the generic element of W , we start with the generic element of P 3 ( R ) and apply the constraints. Define f ( x ) = a + bx + cx 2 + dx 3 . If f W , then 0 = f 0 (0) = ( b + 2 cx + 3 dx 2 ) | x =0 = b Thus W = { a + cx 2 + dx 3 : a, c, d R } . Let β = { 1 , x 2 , x 3 } . Clearly this is a spanning set – given f W , f = a + cx 2 + dx 3 = a (1) + c ( x 2 ) + d ( x 3 ) .
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