Question: (3) (25 pts) Let V' = V3 be the vector space of polynomials in a single variable x of degree at most 3 with complex

(3) (25 pts) Let V' = V3 be the vector space of polynomials in a single variable x of degree at most 3 with complex coefficients. Let 7" be the linear operator T(f) = f + f". (a) Compute the eigenvalues of T. (b) For each eigenvalue, find a basis of the corresponding eigenspace. (c) Give a basis of V for which T is represented by a diagonal matrix. Note that phrased differently, this problem amounts fo finding the values of the parameter A for which the differential equation \\f zf' f" =0 has a polynomial solution of degree 3 or less, and for such values finding the solutions

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