Suppose V is a finite-dimensional with dim V > 1 and T € L(V). Prove that {p(T)|p € F[x]} ‡ L(V). This proof can be done without assuming the T has an eigenvalue. (If the field of scalars is R or Q, then not every operator need have an eigenvalue.)

Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.

Let T: M₂x2 (R) → M2×2(R) be the linear operator given by the trans- pose, i.e. T(A) = A¹, (a) Compute det T, the determinant of T. (b) Compute pr(x), the characteristic polynomial of T. (c) Compute the eigenvalues of T.

4. Let S E L(P3(R), P3(R)) be defined by S(p(x)) = xp'(x). Find a basis for null(S). Is S injective?