Linear Algebra, Theory And Applications, 2012a

364 NORMS FOR FINITE DIMENSIONAL VECTOR SPACES
2. Solve the system ⎛
⎝ 4 1 1
⎞ ⎛
1 7 2 ⎠ ⎝ x ⎞ ⎛
y ⎠ = ⎝ 1 ⎞
2 ⎠
0 2 4 z 3
using the Gauss Seidel method and the Jacobi method. Check your answer by also
solving it using row operations.
3. Solve the system ⎛
⎝
5 1 1
1 7 2
0 2 4
⎞ ⎛
⎠ ⎝
x
y
z
⎞ ⎛
⎠ = ⎝
using the Gauss Seidel method and the Jacobi method. Check your answer by also
solving it using row operations.
4. If you are considering a system of the form Ax = b and A −1 does not exist, will either
the Gauss Seidel or Jacobi methods work? Explain. What does this indicate about
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 + y
∣∣
2 ∣∣ < ||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 ||x n || , ||y n || are equal to 1 for
all n ∈ N and lim n→∞ ||x n + y n || =2, it follows lim n→∞ ||x n − y n || =0. Show the
norm |·| coming from an inner product is always uniformly convex. Also show that
uniform convexity implies strict convexity which is defined in Problem 6.
8. Suppose A : C n → C n is a one to one and onto matrix. Define
Show this is a norm.
||x|| ≡ |Ax| .
9. If X is a finite dimensional normed vector space and A, B ∈ L(X, X) such that
||B|| < ||A|| , can it be concluded that ∣ ∣ ∣ ∣A −1 B ∣ ∣ ∣ ∣ < 1?
10. Let X be a vector space with a norm ||·|| and let V =span(v 1 , ··· ,v m ) be a finite
dimensional subspace of X such that {v 1 , ··· ,v m } is a basis for V. Show V is a closed
subspace of X. This means that if w n → w and each w n ∈ V, then so is w. Next show
that if w/∈ V,
dist (w, V ) ≡ inf {||w − v|| : v ∈ V } > 0
is a continuous function of w and
1
2
3
⎞
⎠
|dist (w, V ) − dist (w 1 ,V)| ≤‖w 1 − w‖
Next show that if w/∈ V, there exists z such that ||z|| =1anddist(z,V ) > 1/2. For
those who know some advanced calculus, show that if X is an infinite dimensional
vector space having norm ||·|| , then the closed unit ball in X cannot be compact.
Thus closed and bounded is never compact in an infinite dimensional normed vector
space.