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∞.
