Math 136
Sample Term Test 1
NOTES:  Questions 4d, 5, 6 on this test cover material that will not be covered
on our term test 1.
 Students had 90 minutes to write this test, where you will have 110 minutes.
1.
Short Answer Problems
a) List the 3 elementary row operations.
b) Does the spanning set span
{
(1
,
2
,
1)
,
(0
,
1
,
0)
,
(1
,
0
,
1)
}
represent a line
or a plane in
R
3
? Give a vector equation which describes it.
c) What is the area of the parallelogram induced by
±a
= (2
,
3) and
±
b
= (1
,

1).
d) If
A
is an
n
×
m
matrix and
B
is an
m
×
p
matrix, then what is the size of
AB
?
e) Explain why
×
(
±
b
×
±
c
) must be a vector in the plane with vector equation
±x
=
s
±
b
+
t±c
,
s, t
∈
R
.
2.
Consider the system of linear equations:
z
1

z
2
+
iz
3
=2
i
(1 +
i
)
z
1

iz
2
+
iz
3
=

2+
i
(1

i
)
z
1
+(

1+2
i
)
z
2
+
(1 + 2
i
)
z
3
=
3 + 2
i
a) Row reduce the matrix to RREF using elementary row operations.
b) What is the rank of the coe±cient matrix?
c) Find the general solution of the system.
3.
Let
S
=
1
2
1
,

1
3
2
,
2
4

1
.
a) Determine if
1
1
1
is in the span of
S
. If so, write it as a linear combination of the
vectors in
S
.
b) Determine if
S
is linearly independent or dependent.
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 Winter '08
 All
 Math, Linear Algebra, Vector Space, 90 minutes, 110 minutes

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