Question 1) Find l_∞ and l₂ norms of the vectors.

x = (sin k, cos k, 2^k)^t for a fixed positive integer k

Question 2) Find l_∞ norm of the matrices.

Question 3) The following linear system Ax = b have x as the actual solution and x˜ as an approximate solution. Compute ||x-x˜||_∞ and ||Ax˜ - b||_∞.

a. 1/2x₁ + 1/2x₂ = 1/63,

    1/3x₁ + 1/4x₂ = 1/168,

    x = (1/7 - 1/6)^t,

    x˜ = (0.142, 0.166)^t.

Question 4) Compute the eigenvalues and associated eigenvectors of the following matrices.

Question 5) Find the complex eigenvalues and associated eigenvectors for following matrices.

Question 6) Find the spectral radius for each matrix in question 5.

Question 7) Find the l₂ norm for the matrices in question 5.

Question 8) Find the first two iterations of the Jacobi method for the following linear system, using x⁽⁰⁾ = 0:

a. 3x₁ - x₂ + x₃ = 1,

    3x₁ + 6x₂ + 2x₃ = 0,

    3x₁ + 3x₂ + 7x₃ = 4.

Question 9) Give an example of the matrix that is convergent.