Let A = (1 1 -5 0 2 -5 0 0 -3), P = (1 1 1 0 1 1 0 0 1), Q = (1 0 1 0 1 1 0 0 1), w = (1 1 0) Let B = {v_1, v_2, v_3} be the basis consisting of the columns of P. Let K = {u_1, u_2, u_3} be the basis consisting of the columns of Q. (a) i. Verify that v_1, v_2, v_3 are eigenvectors of A, and find the corresponding eigenvalues. ii. Hence write down P^-1 AP (you should be able to write this out directly, without multiply it out) (b) Write down the B-coordinates and the K-coordinates of w. (c) Consider the linear transformation T with standard matrix A. That is, T: R^3 rightarrow R^3, T(v) = Av. i. Find [T]_B. ii. Find [T]_K; iii. Find T(w) using A, [T]_B, [T]_K (d) Consider the bilinear form f with standard matrix A. That is. f: R^3 times R^3 R, f(u, v) = u^t Av. i. Find [f]_B, ii. Find [f]_K. iii. Find f(w, w) using A, [f]_B, [f]_K. iv. Is f an inner product?
Show transcribed image text Let A = (1 1 -5 0 2 -5 0 0 -3), P = (1 1 1 0 1 1 0 0 1), Q = (1 0 1 0 1 1 0 0 1), w = (1 1 0) Let B = {v_1, v_2, v_3} be the basis consisting of the columns of P. Let K = {u_1, u_2, u_3} be the basis consisting of the columns of Q. (a) i. Verify that v_1, v_2, v_3 are eigenvectors of A, and find the corresponding eigenvalues. ii. Hence write down P^-1 AP (you should be able to write this out directly, without multiply it out) (b) Write down the B-coordinates and the K-coordinates of w. (c) Consider the linear transformation T with standard matrix A. That is, T: R^3 rightarrow R^3, T(v) = Av. i. Find [T]_B. ii. Find [T]_K; iii. Find T(w) using A, [T]_B, [T]_K (d) Consider the bilinear form f with standard matrix A. That is. f: R^3 times R^3 R, f(u, v) = u^t Av. i. Find [f]_B, ii. Find [f]_K. iii. Find f(w, w) using A, [f]_B, [f]_K. iv. Is f an inner product?