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Math121sol4

Math121sol4 - Math 121 Linear Algebra and Applications...

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Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 4 Posted: Tue. Nov. 6, 2007 Written by: Luca Candelori Exercise 1 (2.1/15) . Linearity follows directly from the properties of integrals. Namely x 0 ( f ( t ) + g ( t )) dt = x 0 f ( t ) dt + x 0 g ( t ) dt It is also 1-1 for suppose that x 0 f ( t ) dt = 0, the zero polynomial. From the Fundamental Theorem of Calculus: d dx x 0 f ( t ) dt = f ( x ) = d dt (0) = 0 hence f ( x ) is the zero function, i.e. N ( T ) = 0 and the map is 1-1. Let now c be a constant in P ( R ). Suppose that there is a f ( x ) in P ( R ) such that x 0 f ( t ) dt = c . By applying derivatives to both sides we get that f ( x ) = 0, which is impossible, since the zero function integrates to 0, and not to c . Therefore there is no such f and the map T is not surjective. Exercise 2 (2.1/20) . Let u 1 , u 2 T ( V 1 ) and c a non zero scalar in F . Then there exists v 1 , v 2 V 1 such that T ( v 1 ) = u 1 and T ( v 2 ) = u 2 . Therefore cu 1 + u 2 = cT ( v 1 ) + T ( v 2 ) = T ( cv 1 + v 2 ) Since V 1 is a subspace, cv 1 + v 2 V 1 and therefore cu 1 + u 2 T ( V 1 ). So T ( V 1 ) is a subspace. Let now S = { x V | T ( x ) W 1 } . We want to show this is a subspace of V . Take x 1 , x 2 S and c a nonzero scalar in F . Then T ( cx 1 + x 2 ) = cT ( x 1 ) + T ( x 2 ) W 1 since T ( x 1 ) , T ( x 2 ) W 1 and W 1 is a subspace of W . Therefore cx 1 + x 2 S .

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Math121sol4 - Math 121 Linear Algebra and Applications...

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