please answer the question

7. Consider the matrix A = 4 .1 (@) Verify that ) =3 is the only eigenvalue of matrix A. (b) Determine a basis for the eigenspace of the matrix A corresponding to the eigenvalue ) = 3. Is A diagonalizable? Explain. (c) Solve the linear system (A - )Ijx = v1, where vi is an eigenvector of the matrix A with eigenvalue * = 3. Choose ve to be a specific solution to this system. (d) Let B = {vi, ve), using the vectors from the previous problems. Verify that B is a basis for RR'. (e) Compute Avy and Ave, using coordinates relative to the basis B. You should do this first by using the relationships between v, and v, and then check your answer by direct computation. (f) Finally, determine the matrix / which represents the transformation T : R' - R', T(x) = Ax with coordinates relative to the basis B