2. The functions f(x) = x sin x, f2(x) = x cos x, f(x) = sin...

 

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2. The functions f(x) = x sin x, f2(x) = x cos x, f(x) = sin x, f(x) = cos x span a 4-dimensional subspace V of the vector space F(R). Let T: V F(R) be linear transformation defined by (Tf)(x) = f(x + 1), x = R, and for any f = F(R). (a) Show that R(T) = V and N(T) = {0}. (b) Compute [T] , where ={ f, f2, f3, f4}. I 2. The functions f(x) = x sin x, f2(x) = x cos x, f(x) = sin x, f(x) = cos x span a 4-dimensional subspace V of the vector space F(R). Let T: V F(R) be linear transformation defined by (Tf)(x) = f(x + 1), x = R, and for any f = F(R). (a) Show that R(T) = V and N(T) = {0}. (b) Compute [T] , where ={ f, f2, f3, f4}. I

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To show that RT V and NT 0 we need to demonstrate two things a RT V We need to show that the range of the linear transformation T denoted as RT is equal to the subspace V spanned by the functions fx x ... View the full answer