June 22, 2006
Final Quiz and Key
Prof. S. Hudson
The unlabelled problems are 10 points each.
1) [15pts] Which of these subspaces ? (answer Yes or No to each one)
{
(
x
1
,x
2
,x
3
)
T

x
1
+
x
2
= 1
}
(in
V
=
R
3
)
{
(
x
1
,x
2
,x
3
)
T

x
1
=
x
2
=
x
3
}
(in
V
=
R
3
)
The set of all 2x2 lower triangular matrices (in
V
=
R
2
x
2
).
The set of all 2x2 singular matrices (in
V
=
R
2
x
2
).
The set of all polynomials in
P
4
of degree 2.
2) Let
S
= span
{
(1
,
2
,
3
,
0)
T
,
(1
,
0
,
0
,
1)
T
}
, a subspace of
R
4
. Find a basis
of
S
⊥
.
3) [20pts] Use the following
XDX

1
factorization to quickly calculate these.
You do not have to simplify very much in a) to d). Numbers like 5
4
are
OK. You shouldn’t have to multiply matrices (but, for example, I would
not accept “
A
2
A
5
” as an answer to part b)
A
=
4
3
3

5
4

5
5

3
6
=
1

1
0
1
0

1

1
1
1
4
0
0
0
1
0
0
0
9
1
1
1
0
1
1
1
0
1
a)
A

1
b)
A
7
c)
e
A
d) det (A)
e) Find a matrix
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Orthogonal matrix, lower triangular matrices, Prof. S. Hudson, little extra credit

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