B U Department of Mathematics
Math 201 Matrix Theory
Spring 2005 First Midterm
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1.) a)
Find
c
such that the following set of columns is a basis for
R
3
:
1
1

1
,
2
1
0
,
1
1
c
.
Solution:
A
=
1
2
1
1
1
1

1
0
c

r
1
+
r
2
→
r
2
→
r
1
+
r
3
→
r
3
1
2
1
0

1
0
0
2
c
+ 1
2
r
2
+
r
3
→
r
3
→
1
2
1
0

1
0
0
0
c
+ 1
. Hence
c
=

1, i.e.,
∀
c
∈
R
\ {
1
}
the given set of columns is a basis for
R
3
.
b)
Is the set of polynomials
S
=
{
1

x,
1 +
x,
1

x
2
}
linearly independent?
Solution:
Consider
a
(1

x
) +
b
(1 +
x
) +
c
(1

x
2
) = 0
Then

cx
2
= 0 implies
c
= 0. So
a
+
b
= 0 and

a
+
b
= 0 give that
a
= 0,
b
= 0. Thus
S
is linearly independent.
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