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Consider the linear transformation T : P 3 (R) P 3 (R) given by T(f (2)) = f' (20) - 2f(20) where f' and f

Consider the linear transformation T : P3(R) P3(R) given by

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T(f (2)) = f' (20) - 2f"(20) where f' and f" denote the first and second derivatives of f. [Do not have to check that T is linear.] (a) Compute the rank of T. (b) Find a basis for the range of T. (c) Find a basis for the null-space of T. (d) Find the eigenvalues of T. (e) Determine whether T is diagonalizable. If it is, find a basis S in which T has a diagonal matrix. Justify your answers

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