Math 416  Practice Exam 1
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Problem 1.
Consider the following ”addition”
⊕
and ”scalar multiplication”
in
R
2
.
x
1
x
2
⊕
y
1
y
2
=
x
1
+
y
1
+ 1
x
2
+
y
2
+ 1
c
x
1
x
2
=
cx
1
+
c

1
cx
2
+
c

1
Verify that
R
2
is a vector space over
R
with respect to these operations!
Problem 2.
Let (
V,
+
,
·
) be a vector space over
R
. Is it possible to make
V
a vector space
over
C
using the same addition and a new scalar multiplication
defined as (
a
+
bi
)
v
:=
a
·
v
?
Problem 3.
Recall that if
A
= (
a
ij
)
2
M
n
⇥
n
(
R
), then the trace tr(
A
) of
A
is the sum of the
diagonal entries of
A
, that is tr(
A
) =
P
n
i
=1
a
ii
. Show that if
A, B
2
M
n
⇥
n
(
R
) and
c
2
R
, then
tr(
cA
+
B
) =
c
tr(
A
) + tr(
B
). Determine if the subset
W
consisting of all trace zero matrices
in
V
= M
n
⇥
n
(
R
) is a subspace of
V
. Then give a basis for
W
and compute the dimension of
W
. Begin with the case of
n
= 2.
Problem 4.
Let
X
and
Y
be subspaces of a vector space
V
.
Show that with respect to
inclusion,
1)
X
\
Y
is the largest subspace of
V
contained in both
X
and
Y
.
2)
X
+
Y
is the smallest subspace of
V
containing both
X
and
Y
.
Problem 5.
Let
V
=
M
2
⇥
2
(
R
) regarded as a vector space over
R
. For the following subsets
X, Y
of
V
, determine if they are subspaces. Then describe
X
\
Y
and
X
+
Y
by giving a basis
for
X
,
Y
,
X
\
Y
,
X
+
Y
and computing their dimensions.
1.
X
=
{
A
2
V

tr(
A
) = 0
}
, and
Y
= diagonal matrices in
V
.
2.
X
=
⇢
0
0
c
d

c, d
2
R
, and
Y
=
⇢
0
a
b
0

a, b
2
R
Problem 6.
Let
X
and
Y
be subspaces of a vector space
V
. Prove that if
X
\
Y
=
{
0
}
then
dim(
X
+
Y
) = dim(
X
) + dim(
Y
) What about the case that
X
\
Y
6
=
{
0
}
?
Problem 7.
Is it possible that vectors
v
1
, v
2
, v
3
are linearly dependent, but the vectors
v
1

v
2
, v
2

v
3
, v
3

v
1
are linearly independent?
1
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Problem 8.
Let
a
1
, a
2
, a
3
be distinct numbers in
R
and consider the following polynomials:
p
1
(
x
) = (
x

a
2
)(
x

a
3
)
, p
2
(
x
) = (
x

a
1
)(
x

a
3
)
, p
3
(
x
) = (
x

a
1
)(
x

a
2
)
Compute dim
R
(span
{
p
1
, p
2
, p
3
}
).
Problem 9.
What is the dimension of the span of
S
=
{
sin
x,
cos
x,
sin 2
x,
cos 2
x
}
⇢
C
(
R
,
R
)
over
R
?
Problem 10.
Determine if the following subsets are subspaces of
R
3
. Then give a basis and
compute the dimension. Also describe them geometrically.
Problem 11.
Let
V
= P
2
(
R
) and consider the following subsets
W
of
V
. Determine if
W
is
a subspace of
V
, give a basis for
W