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MAS 3105
Feb 25, 2004
Quiz 4 Key
Prof. S. Hudson
1) The matrix
A
is row equivalent to
U
(see below). Find a basis for
N
(
A
), and explain
brieﬂy how you know it is a basis.
A
=
1
3
0
9
4
1
1
4
19
2
1
1
2
11
2
U
=
1
0
0
0
1
0
1
0
3
1
0
0
1
4
0
2) Which of the following sets
S
are subspaces of
R
2
? Explain each answer brieﬂy.
a)
S
=
{
(
x
1
,x
2
)
T

3
x
1
+ 4
x
2
= 7
}
b)
S
=
{
(
x
1
,x
2
)
T

x
1
=
x
2
}
c)
S
=
{
(
x
1
,x
2
)
T

x
1
=
x
1
}
d)
S
=
{
(
x
1
,x
2
)
T

x
2
1
+
x
2
2
= 0
}
3) Choose ONE of these to prove. You can answer on the back.
a) If dim(V)=
n >
0 and
B
=
{
v
1
,v
2
,...,v
n
}
spans
V
then
B
is a basis of V.
b) Show that a nonempty subset of a linearly independent set of vectors
{
v
1
,v
2
,...v
n
}
is also linearly independent.
c) Let
L
=
{
v
1
,v
2
,...,v
n
}
be a spanning set of
V
and let
v
∈
V
be any other vector.
Show that
{
v
1
,v
2
,...,v
n
,v
}
is linearly dependent.
Extra Credit (about 35 points): Should a Linear Algebra course include MATLAB in the
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 Spring '09
 JULIANEDWARDS

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