We consider the approximations to the eigenvalues and eigenfunctions of the one-dimensional Laplace Operator L(u) = -d62 /dx^2 on the unit interval [0, 1] with boundary u(0) = u(1) = 0. A scalar lambda is an eigenvalue of L if there exists a twice -differentiable function u s. t. -u"(x) = lambda u(x) on [0, 1] with u(0) = u(1) = 0. In this case u is said to be an eigenfunction of L corresponding to the eigenvalue lambda. This continuous problem can be approximated by the discrete eigenvalue problem h^-2 T_N u = lambda u Where we have set T = [2 -1 0 -1 -1 0 -1 2], u = [u_1 u_2 u_n], with u_i = u(x_i). It can be shown that the N times N matrix T_N has eigenvalues v_j = 2(1 - cos xj/N + 1) for j = 1: N, corresponding to the eigenvectors u_j (k) = squareroot 2/N + 1 sin (jk pi/N + 1) is the kth entry in u_j. Notice that the eigenvectors are normalized w.r.t. the 2-norm. Also notice that eigenvalues of T_n lie in the interval (0, 4). Hence, the eigenvalues of h^-2 T_N lie in the interval (0, 4(N + 1)^2). (a) Express the spectral condition number kappa = lambda_N/lambda_1 of T_N (which is the same as the condition number of h^-2 T_N) as a simple function of N for N rightarrow infinity. Describe the conditioning of the T_N. (b) Use Matlab command eig to find the eigenvalues of T_N for N = 5. (c) Use Matlab to plot the eigenvalues of T_N for N = 21 (from smallest to largest). (d) Again using N = 21, use Matlab to plot the eigenvectors u_j of T_N for j = 1, 2, 3, 5, 11, 21. The plot should be of the form (k, u_j (k)) where k = 1: 21. Note that as j increases, the behavior of the eigenvectors (approximate eigenfunctions) becomes increasingly oscillatory: these eigenfunctions are often called "high energy modes