Math 136
Sample Term Test 2 # 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) What is the definition of the row space and column space of a matrix
A
.
Solution: The row space is the spanning set of the rows of
A
and the column space is the
spanning set of the columns of
A
.
b) What is the definition of a subspace.
Solution: A subspace is a subset of a vector space
V
that is also a vector space under the
same operations as
V
.
c) What is the definition of a basis.
Solution: A linearly independent spanning set.
d) Let
B
=
{
(1
,
2
,
1)
,
(

1
,
0
,
2)
,
(1
,
1
,
1)
}
. If [
v
]
B
=
1

1
1
what is
v
?
Solution:
v
= 1(1
,
2
,
1) + (

1)(

1
,
0
,
2) + (1)(1
,
1
,
1) = (3
,
3
,
0)
.
e) Let
V
be a vector space and
v
∈
V
. Prove that (

1)
v
is the additive inverse of
v
.
Solution: We have
v
+ (

1)
v
= (1 + (

1))
v
= 0
v
= 0
,
by V8. Thus (

1)
v
is the additive inverse.
2.
Let
β
=
{
1 +
x
2
,
1 +
x
+
x
2
,
1 + 2
x
+ 2
x
2
}
.
a) Show that
β
is a basis for
P
2
.
Solution: Consider
c
1
(1 +
x
2
) +
c
2
(1 +
x
+
x
2
) +
c
3
(1 + 2
x
+ 2
x
2
) = 0. Rowreducing the
coe
ffi
cient matrix gives
1
1
1
0
1
2
1
1
2
∼
1
1
1
0
1
2
0
0
1
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 All
 Math, Linear Algebra

Click to edit the document details