Math 235
Assignment 3 Solutions
1.
For each of the following pairs of vector spaces, define an explicit isomorphism to
establish that the spaces are isomorphic. Prove that your map is an isomorphism.
a)
R
4
and
M
(2
,
2).
Solution: We define
L
:
R
4
→
M
(2
,
2) by
L
(
a
1
, a
2
, a
3
, a
4
) =
a
1
a
2
a
3
a
4
.
To prove that it is an isomorphism, we must prove that it is linear, onetoone and onto.
Linear: Let any two elements of
R
4
be
a
= (
a
1
, a
2
, a
3
, a
4
) and
b
= (
b
1
, b
2
, b
3
, b
4
) and let
k
∈
R
then
L
(
ka
+
b
) =
L
(
k
((
a
1
, a
2
, a
3
, a
4
) + (
b
1
, b
2
, b
3
, b
4
)
)
=
L
(
ka
1
+
b
1
, ka
2
b
2
, ka
3
+
b
3
, ka
4
+
b
4
)
=
ka
1
+
b
1
ka
2
+
b
2
ka
3
+
b
3
ka
4
+
b
4
=
k
a
1
a
2
a
3
a
4
+
b
1
b
2
b
3
b
4
=
kL
(
a
) +
L
(
b
)
Therefore
L
is linear.
Onetoone: Assume
L
(
a
) =
L
(
b
). Then
L
(
a
1
, a
2
, a
3
, a
4
) =
L
((
b
1
, b
2
, b
3
, b
4
)
⇒
a
1
a
2
a
3
a
4
=
b
1
b
2
b
3
b
4
.
This gives
a
1
=
b
1
,
a
2
=
b
2
,
a
3
=
b
3
,
a
4
=
b
4
hence
a
=
b
so
L
is onetoone.
Onto: For any
a
1
a
2
a
3
a
4
∈
M
(2
,
2) we have
L
(
a
1
, a
2
, a
3
, a
4
) =
a
1
a
2
a
3
a
4
hence
L
is onto.
Thus,
L
is an isomorphism from
R
4
to
M
(2
,
2).
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2
b)
P
2
and
M
(3
,
1).
Solution: We define
L
:
P
2
3
→
M
(3
,
1) by
L
(
a
0
x
2
+
a
1
x
+
a
2
) =
a
0
a
1
a
2
.
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 Fall '08
 CELMIN
 Linear Algebra, Vector Space, Isomorphism, Howard Staunton

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