Problem 2. TRUE/FALSE. (3 points each) Be sure to clearly state whether the items below are

true or false. Then you must fully justify your answer: if true, give a brief explanation (e.g. cite a

relevant theorem from the book or class notes, or show the necessary calculations); if false, supply

a counterexample.

(a) If an n x n matrix A is diagonalizable, then so is A'.

(b) If v is an eigenvector of an n x n matrix A with corresponding eigenvalue ), then v is also an

eigenvector of the matrix A - aln, for every de R.

(c) Given a vector space V, the change of coordinates matrix [Iv] from #' to 8 is always diagonal-

izable.

(d) The linear operator 7 : R3 -> R3 that reflects a point (r, y, z) across

the plane ar + by +c= =0 (a,b,ce R) is diagonalizable.