Math 136
Sample Term Test 2 # 2 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) Let
S
=
{
v
1
, .. . ,v
n
}
be a nonempty subset of a vector space
V
. Deﬁne the statement
S
is linearly independent.
Solution:
S
is linearly independent if the only solution of
c
1
v
1
+
c
2
v
2
+
···
+
c
n
v
n
= 0 is
c
1
=
c
2
=
···
=
c
n
= 0.
b) Write the deﬁnition of a subspace
S
of a vector space
V
.
Solution:
S
is a subspace of
V
if
S
is a subset of
V
and
S
is a vector space using the same
operations as
V
.
c) Write the deﬁnition of the dimension of a vector space
V
.
Solution: The dimension of
V
is the number of elements in any basis for
V
.
d) Prove that 0
x
=
0
for any
x
∈
V
.
Solution: By vector space axioms 3,4,2,8 and 10, 0
x
= 0
x
+ 0 = 0
x
+ (
x
+ (

x
)) =
(0
x
+
x
) + (

x
) = (0 + 1)
x
+ (

x
) = 1
x
+ (

x
) =
x
+ (

x
) = 0.
e) Is it true that if a set
S
with more than one vector is linearly dependent then every vector
v
∈
S
can be written as a linear combination of the other vectors. Justify your answer.
Solution: It is false.
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.
 Spring '08
 All
 Math, Linear Algebra, Algebra, Vector Space

Click to edit the document details