B U Department of Mathematics
Math 201 Matrix Theory
Spring 2002 Final Exam
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1.
Consider in
R
3
the vectors:
u
=
1
1
0
,
v
=
2
2
1
, w
=
0
0
2
, and
b
=
3
4
5
.
(a)
Determine whether
{
u, v, w
}
is a linearly independent set.
(b)
Determine whether
b
is in the subspace spanned by
{
u, v, w
}
.
Solution:
(a)
Clearly not linearly independent, since
v
= 2
u
+
1
2
w
.
(b)
Consider
A
.
.
.
b
=
1
2
0
.
.
.
3
1
2
0
.
.
.
4
0
1
2
.
.
.
5
→
1
2
0
.
.
.
3
0
0
0
.
.
.
1
0
1
2
.
.
.
5
→
1
2
0
.
.
.
3
0
1
2
.
.
.
5
0
0
0
.
.
.
1
so
Rank
(
A
.
.
.
b
) = 3 and
Rank
(
A
) = 2.
⇒
b
is not in the
Columnspace
(
A
). Therefore,
b
is not in the subspace of
R
3
which is spanned by
{
u, v, w
}
.
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 Spring '08
 sadik
 Math, Linear Algebra, UK, yk, Det, Boazi¸i University Mathematics Department

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