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(a) Prove that if two linear transformations T,U : V --> W have the same values...

(a) Prove that if two linear transformations T,U : V --> W have the same values on a basis for V, i.e., T(x) = U(x) for all x belong to beta , then T = U. Conclude that every linear transformation is uniquely determined by the images of basis vectors.

(b) (7 points) Determine the linear transformation T : P1(R) --> P2(R) given by T (1 + x) = 1+x^2, T(1- x) = x by finding the image T(a+bx) of a+bx for a+bx belong to P1(R). Here, {1+x,1 x} is a basis for P1(R).

(c) (3 points) Is the linear map T from part b. injective? Explain your answer.

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