14.5. EXERCISES 403

14.5 Exercises1. Solve the system  4 1 1

1 5 20 2 6

 x

yz

=

 123

using the Gauss Seidel method and the Jacobi method. Check your answer by alsosolving it using row operations.

2. Solve the system  4 1 11 7 20 2 4

 x

yz

=

 123

using the Gauss Seidel method and the Jacobi method. Check your answer by alsosolving it using row operations.

3. Solve the system  5 1 11 7 20 2 4

 x

yz

=

 123

using the Gauss Seidel method and the Jacobi method. Check your answer by alsosolving it using row operations.

4. If you are considering a system of the form Ax= b and A−1 does not exist, willeither the Gauss Seidel or Jacobi methods work? Explain. What does this indicateabout finding eigenvectors for a given eigenvalue?

5. For ||x||∞≡ max

{∣∣x j∣∣ : j = 1,2, · · · ,n

}, the parallelogram identity does not hold.

Explain.

6. A norm ||·|| is said to be strictly convex if whenever ||x||= ||y|| ,x ̸= y, it follows∣∣∣∣∣∣∣∣x+ y2

∣∣∣∣∣∣∣∣< ||x||= ||y|| .Show the norm |·| which comes from an inner product is strictly convex.

7. A norm ||·|| is said to be uniformly convex if whenever ||xn|| , ||yn|| are equal to 1for all n ∈ N and limn→∞ ||xn + yn|| = 2, it follows limn→∞ ||xn− yn|| = 0. Show thenorm |·| coming from an inner product is always uniformly convex. Also show thatuniform convexity implies strict convexity which is defined in Problem 6.

8. Suppose A : Cn→ Cn is a one to one and onto matrix. Define

||x|| ≡ |Ax| .

Show this is a norm.