Linear Algebra Homework 1 Selected Solutions
Any mistakes are the sole responsibility of the authorT.Peters
1.1.6g
Represent as a matrix and row reduce:
1
3
2
3
2

1
1
2
3
2
3
2
1
2
2
12
5
1
10
1
2
6

3
2
4
3
3
5
20
24
1
1
2
6

3
0
0

9
9
0
10

6
16
1
2
6

3
0
0
1

1
0
5

3
8
1
2
6

3
0
1
0
1
0
0
1

1
1
2
0
3
0
1
0
1
0
0
1

1
1
0
0
1
0
1
0
1
0
0
1

1
So
x
1
=

1
, x
2
=

1
,
and
x
3
= 1.
1.1.9
This whole problem can be interpreted geometrically: we have two lines
l
1
and
l
2
defined by
the equations. The system has a unique solution if and only if the two lines are not parallel,
ie
m
1
=
m
2
. If
m
1
=
m
2
then the only way for the system to be consistent is for the lines to
be coincident, ie be the same line. This happens only when
b
1
=
b
2
.
1.2.8b
Represent as a matrix and rowreduce:
1
3
1
1
3
2

2
1
2
8
3
1
2

1

1
1
3
1
1
3
0

8

1
0
2
0

8

1

4

4
1
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1
3
1
1
3
0
1
1
8
0

1
4
0
0
0
1
3
1
0
5
8
0
0
0
1
1
8
0

1
4
0
0
0
1
3
So that
x
1
=

5
8
t
,
x
2
=

1
4

1
8
t
,
x
3
=
t
, and
x
4
= 3.
1.2.9
(a) It is not possible for the system to be inconsistent since this only happens when then
RREF of the matrix has a row of the form(0 0 0

1) but this is not possible since the 4th
column is all zeros–and this property will be preserved under row operations. Alternatively,
we could realize that this is a homogeneous system, and homogeneous systems are always
consistent since they have the trivial solution.
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 Spring '10
 Peters
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