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 3,4,2,8,10 and 4, 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.
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 Spring '08
 All
 Math, Linear Algebra, Algebra, Vector Space

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