Math 20F
Midterm Exam (version 1)
April 29, 2005
1. (6 points) Consider the system
x
1

2
x
2
+
2
x
3
=
10
x
2
+
3
x
3
=
6
x
1

3
x
2

x
3
=
4
.
(a) Determine the solution set of the system and write it in parametric form.
The augmented matrix is
B
=
1

2
2
10
0
1
3
6
1

3

1
4
.
The reduced echelon form of
B
is
1
0
8
22
0
1
3
6
0
0
0
0
.
Thus,
x
3
is free,
x
1
=

8
x
3
+ 22, and
x
2
=

3
x
3
+ 6.
The parametric form of the solution
set is
x
=
t

8

3
1
+
22
6
0
, t
any scalar
.
(b) The coefficient matrix for the above system is
A
=
1

2
2
0
1
3
1

3

1
.
Are the columns of
A
linearly independent or linearly dependent? Justify your
answer.
The columns of
A
are linearly
dependent
since the reduced echelon form of
A
has a zero row and
therefore the homogeneous equation
A
x
=
0
has nontrivial solutions. In fact,
8
1
0
1
+ 3

2
1

3

2
3

1
=
0
0
0
is a typical dependence relation.
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2. (8 points) A linear transformation
T
:
2
→
3
satisfies
T
(
e
1
) = (2
,
1
,
1)
and
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 Spring '03
 BUSS
 Math, Linear Algebra, Algebra, standard matrix

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