M371 - Alexiades
Problem Set 4: Linear Algebra (do by hand)
1. Consider the linear system
4x1 + x2 = 2
x1 + 5x2 = 10
a. 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 A-1.
e. Solve the system via x = A-1b.
2. Let A = the coefficient matrix in Problem 1, and b = [2,10]T.
a. Find the 1-norm, the 2-norm, and the infinity-norm of b.
b. Find the 1-norm and the infinity-norm of A.
c. Find the 1-norm and the infinity-norm of A-1.
3. The exact solution in Problem 1 is x = [0 2]T.
Considering y = [0.1 2.1]T as an approximation to 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 number κ(A) in inf-norm.
d. Verify the absolute error bound: ∥e∥ ≤ ∥A-1∥∥r∥ (in inf-norm).
e. Verify the relative error bound: ∥e∥/∥x∥ ≤ κ ∥r∥/∥b∥ (in inf-norm).
4. For square matrices A, consider the quantity ∥A∥o = max |aij|
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 NOT a good norm! (and it is NOT an induced norm).