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Math 136
Sample Term Test 1  2
NOTE
:  Only answers are provided here (and some proofs). On the test you
must
provide
full and complete solutions to receive full marks.
1.
Short Answer Problems
a) List the 3 elementary row operations.
Solution: 1. Multiply a row by a nonzero constant
2. Swap two rows
3. Add a multiple of one row to another.
b) What can you say about the consistency and the number of parameters (free variables)
in the general solution of a system of 5 linear equations in 4 variables.
Solution: You can not say anything about the consistency or the number of parameters
because the system has more equation than variables and we don’t know the rank of the
coeFcient matrix.
c) What is the area of the parallelogram induced by
±a
= (1
,

2) and
±
b
= (4
,

9).
Solution: Area=
±
±
±
±
det
²
1

2
4

9
³ ±
±
±
±
=

1(

9)

(

2)(4)

= 1.
d) Let
A
=
²
3
2 1

2 1 4
³
and
B
=

21
11
0

1
. Calculate
AB
.
Solution:
AB
=
²

44
5

5
³
.
e) Let
S
=
{
±v
1
,±v
2
3
}
be a set of vectors in
R
3
. State the de±nition of the set
S
being
linearly independent.
Solution:
S
is linearly independent if the only solution to
c
1
1
+
c
2
2
+
c
3
3
=
±
0 is the trivial
solution (
c
1
=
c
2
=
c
3
= 0).
f) Explain why
×
(
±
b
×
±
c
) must be a vector in the plane with vector equation
±x
=
s
±
b
+
t±c
,
s, t
∈
R
.
Solution: Suppose that
±n
=
±
b
×
±
c
±
=
±
0. Then
is orthogonal to both
±
b
and
±
c
, so it is a
normal vector to the plane through the origin that contain
±
b
and
±
c
. Then
×
(
±
b
×
±
c
)=
×
is orthogonal to
so it lies in the plane with normal
, that is, in the plane containing
±
b
and
±
c
. Hence, for some real numbers
s, t
,
a
×
(
±
b
×
±
c
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 Winter '08
 All
 Math

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