## Description

1. Answer the following questions. Justify your answers.

a) Are the columns of the following matrix linearly independent?

A =

+0.92 +0.92

−0.92 +0.92

+0.92 −0.92

−0.92 −0.92

b) Are the columns of the following matrix linearly independent?

A =

+1 +1 +1

−1 +1 −1

+1 −1 −1

c) Are the columns of the following matrix linearly independent?

A =

1 2 2

3 4 5

5 6 8

d) What is the rank of the following matrix?

A =

+5 +2

−5 +2

+5 −2

e) Suppose the matrix in part d is used in to solve the system of linear equations

ATAw = d. Does a unique solution exist? Explain why.

2. Norm additivity. Suppose k·ka

and k·kb

are norms on R

n

.

a) Prove that f(x) = kxka + kxkb

is also a norm on R

n

.

b) The “norm ball” is defined as the set of x for which an (arbitrary) norm f(x) = 1.

Sketch the norm ball in R

2

for the norm f(x) = kxk1 + kxk∞.

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