MAS 3105
April 28, 2011
Final Exam and Key
Prof. S. Hudson
1) Suppose
L
rotates each vector in
R
2
by 45 degrees clockwise. Find the matrix repre
sentation of
L
(standard basis). Hint: cos(45) =
p
1
/
2.
2) [20 pts] TrueFalse. You can assume the matrices are all square.
All the eigenvectors of a nilpotent matrix are 0.
If eigenvectors
x
1
and
x
2
correspond to
λ
1
6
=
λ
2
then
x
1
⊥
x
2
.
If
A
is similar to
B
then they have the same rank.
The Google PageRank algorithm is a Markov process.
∃
A
∈
R
3
×
3
such that (3
,
1
,
1)
∈
Row(
A
) and (1
,
1
,
3)
T
∈
N
(
A
).
If
L
:
V
→
W
is linear, then ker(
L
)
⊥
L
(
V
).
We used a basis of 3 eigenvectors to solve the Rabbit problem.
For all 3
×
3 matrices, rank (
AB
)
≤
rank (
B
).
The normal equations are
AA
T
x
=
A
T
b
.
Overdetermined systems are usually inconsistent.
3) Choose ONE [from Ch 6.4 HW]
a) Show that if
U
is unitary and
x
∈
C
n
, then

U
x

=

x

.
b) Show that if
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Matrices, Diagonalizable matrix, Orthogonal matrix, Unitary matrix, Hermitian

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