Math 136
Sample Term Test 2 # 2
NOTES:  In addition to these questions you should also do questions 7, 10 b from
sample term test 1 # 2.
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.
b) Write the definition of a subspace
S
of a vector space
V
.
c) Write the definition of the dimension of a vector space
V
.
d) Prove that 0
x
=
0
for any
x
∈
V
.
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.
2.
Let
β
=
{
x
2

4
x
+ 4
, x

2
,
1
}
.
a) Show that span(
β
) =
P
2
.
b) Let
w
=
x
2
+
x
+ 1. Find the
β
coordinate vector of
w
.
3.
Determine, with proof, which of the following are subspaces of the given vector space.
a)
S
=
{
ax
2
+
bx
+
c

b
2

4
ac
= 0
}
of
P
2
.
b)
D
=
a
b
c
d
ad

bc
= 0
of
M
(2
,
2).
c)
A
=
a
b
c
d
a
+
b
+
c
+
d
= 0
of
M
(2
,
2).
4.
Let
A
=
3
6
1
1
2
4
2
4
3
and let
L
be a linear mapping with matrix
A
.
a) Find a basis for the nullspace of
L
.
b) Find a basis for the range of
L
.
5.
Find a basis for the following subspaces of
R
3
and state the dimension of the subspace.
a) span
{
(2
,
4
,
6)
,
(4
,
5
,
6)
,
(1
,
1
,
1)
}
b) The plane 2
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 Winter '08
 All
 Linear Algebra, Algebra, Addition, Vector Space, #, Rn Rm, sample term test

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