MAS 3105
April 19, 2004
Final Exam and Key
Prof. S. Hudson
1) (10pts) Suppose that
A
is a square singular matrix. What can you say about the product
A
adj
A
?
2) (10 pts) a) Find the least squares solution to
A
x
=
b
, using this
QR
factorization:
A
=
1

2
1
2
0
1
2

4
2
4
0
0
=
1
5
1

2

4
2
1
2
2

4
2
4
2

1
5

2
1
0
4

1
0
0
2
where
b
=

1
1
1

2
2b) Let
S
=
R
(
A
). Find proj
S
b
.
3) (15pts) a) Given that
A
is row equivalent to
U
, find a basis for
N
(
A
). Check !
A
=
1

2
1
1
2

1
3
0
2

2
0
1
1
3
4
1
2
5
13
5
U
=
1

2
1
1
2
0
1
1
3
0
0
0
0
0
1
0
0
0
0
0
3b) Find a basis of the row space of
A
.
3c) Find a basis of the column space of
A
.
4) (10pts) Find the standard matrix representation for each of the following linear trans
formations,
a)
L
rotates each vector
x
in
R
2
by 45
◦
in the clockwise direction.
b)
L
reflects each vector about the line
x
1
=
x
2
, and then projects it onto the
x
1
axis.
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Matrices, Hermitian, Hermitian matrices, similar square matrices

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