Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample.
(a) A square matrix with linearly independent column vectors is diagonalizable.
(b) If A is diagonalizable, then there is a unique matrix P such that p-1AP is a diagonal matrix.
(c) If v1, v2, and v3 come from different eigenspaces of A, then it is impossible to express v3 as a linear combination of v1 and v2.
(d) If A is diagonalizable and invertible, then a-1 is diagonalizable.
(e) If A is diagonalizable, then ATis diagonalizable.