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)
P
3
and
R
4
.
Solution: We deﬁne
L
:
P
3
→
R
4
by
L
(
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
) = (
a
3
,a
2
,a
1
,a
0
).
Linear:
Let any two elements of
P
3
be
~a
=
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
and
~
b
=
b
3
x
3
+
b
2
x
2
+
b
1
x
+
b
0
and let
k
∈
R
then
L
(
k~a
+
~
b
) =
L
(
k
(
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
) + (
b
3
x
3
+
b
2
x
2
+
b
1
x
+
b
0
)
)
=
L
(
(
ka
3
+
b
3
)
x
3
+ (
ka
2
+
b
2
)
x
2
+ (
ka
1
+
b
1
)
x
+
ka
0
+
b
0
)
=
(
ka
3
+
b
3
,ka
2
+
b
2
,ka
1
+
b
1
,ka
0
+
b
0
)
=
k
(
a
3
,a
2
,a
1
,a
0
) + (
b
3
,b
2
,b
1
,b
0
) =
kL
(
~a
) +
L
(
~
b
)
Therefore
L
is linear.
One-to-one:
Assume
L
(
~a
) =
L
(
~
b
). Then
L
(
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
) =
L
(
b
3
x
3
+
b
2
x
2
+
b
1
x
+
b
0
)
⇒
(
a
3
,a
2
,a
1
,a
0
) = (
b
3
,b
2
,b
1
,b
0
)
.
This gives
a
3
=
b
3
,
a
2
=
b
2
,
a
1
=
b
1
,
a
0
=
b
0
hence
~a
=
~
b
so
L
is one-to-one.
Onto:
For any (
a
3
,a
2
,a
1
,a
0
)
∈
R
4
we have
L
(
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
) = (
a
3
,a
2
,a
1
,a
0
) hence
L
is onto.
Thus,