Math 416  Abstract Linear Algebra
Fall 2011, section E1
Midterm 1, September 21
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Solutions
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1.
/10
2.
/10
3.
/10
4.
/10
Total:
/40
1
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View Full DocumentProblem 1a. (8 pts)
Find all solutions (if any) of the system
x
1
+ 2
x
2

x
3
+ 2
x
4
= 3
3
x
1
+ 7
x
2
+ 5
x
4
= 8

x
1
+ 7
x
3

2
x
4
=

1
.
1
2

1
2
3
3
7
0
5
8

1 0
7

2

1
∼
1 2

1
2
3
0 1
3

1

1
0 2
6
0
2
∼
1 2

1
2
3
0 1
3

1

1
0 0
0
2
4
∼
1 2

1
2
3
0 1
3

1

1
0 0
0
1
2
∼
1 2

1 0

1
0 1
3
0
1
0 0
0
1
2
∼
1 0

7 0

3
0 1
3
0
1
0 0
0
1
2
The only free variable is
x
3
. The solution set is
{

3 + 7
x
3
1

3
x
3
x
3
2
∈
R
4

x
3
∈
R
}
=
{

3
1
0
2
+
x
3
7

3
1
0
∈
R
4

x
3
∈
R
}
.
b. (2 pts)
If we changed the righthand side
3
8

1
to some other vector
b
∈
R
3
, would the
system necessarily have a solution? If so, justify; if not, ﬁnd an explicit righthand side
b
∈
R
3
such that the system has no solution.
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 Fall '11
 FRANKLAND
 Math, Linear Algebra, Algebra, Linear Independence, linearly independent columns

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