Problem Set 4: Linear Algebra (do by hand)

1. Consider the linear system4xa. Show that the coefficient matrix A is non-singular. b. Solve the system by Gauss elimination (REF and back substitution). c. Solve the system by LU factorization of A. d. Use the LU factorization to find_{1}+ x_{2}= 2 x_{1}+ 5x_{2}= 10A. e. Solve the system via^{-1}x = A. 2. Let^{-1}bA= the coefficient matrix in Problem 1, andb = [2,10]. a. Find the 1-norm, the 2-norm, and the infinity-norm of^{T}b. b. Find the 1-norm and the infinity-norm ofA. c. Find the 1-norm and the infinity-norm ofA. 3. The exact solution in Problem 1 is^{-1}x = [0 2]. Considering^{T}y = [0.1 2.1]as an approximation to^{T}x: a. Find the absolute and relative error (e=y-x) in the 1-norm and the inf-norm. b. Find the absolute and relative residual (r=Ay-b) in 1-norm and inf-norm. c. Compute the condition numberκ(A)in inf-norm. d. Verify the absolute error bound:∥e∥ ≤ ∥A(in inf-norm). e. Verify the relative error bound:^{-1}∥∥r∥∥e∥/∥x∥ ≤ κ ∥r∥/∥b∥(in inf-norm). 4. For square matrices A, consider the quantity∥A∥a. Show it is a norm (satisfies the 3 axioms for a norm). b. Show that it does NOT necessarily satisfy the very desirable property ∥AB∥ ≤ ∥A∥∥B∥ (find specific A, B for which it fails). so this is_{o}= max |a_{ij}|! (and it is NOT an induced norm).NOT a good norm