Math 136
Term Test 1 Answers
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) 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.
Solution: This is a plane with vector equation
~x
=
t
(1
,
0
,
1) +
s
(0
,
1
,
0)
.
c) What is the area of the parallelogram induced by
~a
= (2
,
3) and
~
b
= (1
,

1).
Solution: The area is
±
±
±
±
det
²
2
3
1

1
³±
±
±
±
=

2(

1)

3(1)

=
 
5

= 5.
d) If
A
is an
n
×
m
matrix and
B
is an
m
×
p
matrix, then what is the size of
AB
?.
Solution:
AB
is
n
×
p
.
e) Explain why
~a
×
(
~
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
6
=
~
0. Then
~n
is orthogonal to both
~
b
and
~
c
, so it is a
normal vector to the plane through the origin that contain
~
b
and
~
c
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 Winter '08
 All
 Math, Linear Algebra, Algebra, Vector Space, #

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