Page 2 Problems 1. Let Pn be the vector space of polynomials with real coefficients and degree at most n. For each of the following functions, determine if the function F is a linear transformation. If the function F is a linear transformation, determine the kernel and image of F. Is F one-to-one, onto, both, or neither? Recall that a linear transformation T : V - W is one-to-one if and only if Ker T = {0v }, and T is onto if and only if Img T = W. (a) F : P2 - P2 defined by F(p(x) ) = x + p(x). (b) F : P2 - P3 defined by F(p(x) ) = xp(x). (c) F : P2 - P2 defined by F(p(x) ) = p(1 - x). 2. Let C be the vector space of continuous functions f : R - R. Let I = fet, e2x, 3x and V = Span(I). Define a linear transformation T : V - C by T(ex) = 1, T(e2x) = cos2 x, T(e3x ) = sin2 x. (a Write down a formula to compute T(Clet + Cze2x + Cge3x) for any real C1, C2, C3. (b) Show that T(I) is linearly dependent. Using this fact, is T one-to-one? If not, give a non-zero vector in Ker T. (c) Show that T(I) does not span C. Using this fact, is T onto? 1 3. Let T : V - W be a linear transformation. (a) Show that T is one-to-one if and only if for every linearly independent subset I of V, T(I) is linearly independent. (b) Let S. C V be a spanning set. Show that T is onto if and only if T(S) is a spanning set for W