1 (c) prove that, if a and b are similar n n matrices then det a = det b: [5 points] use cramer"s rule to solve the linear system. = 3 z = 2: [8 points] let s = be a subset of r4. 3: [6 points] (a) carefully de ne what it means for the subset {v1, v2 . , vk} of rn to be linearly dependent and linearly independent. (b) let u, v and w be independent column vectors in rn and let a be an invertible n n matrix. Prove that the vectors a u, a v and a w are also independent: [6 points] Justify your answer. (b) for each eigenvalue determine an eigenvector which corresponds to that eigenvalue. Justify your answer: [13 points] a = . Explain your answer. (d) if a is diagonalizable, give matrices c and d such that d = c 1 a c: [13 points] a = .
