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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)
M
(2
,
2) and
P
3
.
Solution: We deﬁne
L
:
M
(2
,
2)
→
P
3
by
L
±²
a b
c d
³´
=
ax
3
+
bx
2
+
cx
+
d
.
To prove that it is an isomorphism, we must prove that it is linear, onetoone and onto.
Linear: Let any two elements of
M
(2
,
2) be
~a
=
²
a
1
b
1
c
1
d
1
³
and
~
b
=
²
a
2
b
2
c
2
d
2
³
and let
k
∈
R
then
L
(
k~a
+
~
b
) =
L
(
k
²
a
1
b
1
c
1
d
1
³
+
²
a
2
b
2
c
2
d
2
³
)
=
L
(
²
ka
1
+
a
2
kb
1
+
b
2
kc
1
+
c
2
kd
1
+
d
2
³
)
=
(
(
ka
1
+
a
2
)
x
3
+ (
kb
1
+
b
2
)
x
2
+ (
kc
1
+
c
2
)
x
+ (
kd
1
+
d
)
)
=
k
(
a
1
x
3
+
b
1
x
2
+
c
1
x
+
d
1
) +
a
2
x
3
+
b
2
x
2
+
c
2
x
+
d
2
=
kL
(
~a
) +
L
(
~
b
)
Therefore
L
is linear.
Onetoone: Assume
L
(
~a
) =
L
(
~
b
). Then
L
±²
a
1
b
1
c
1
d
1
³´
=
L
±²
a
2
b
2
c
2
d
2
³´
⇒
a
1
x
3
+
b
1
x
2
+
c
1
x
+
d
1
=
a
2
x
3
+
b
2
x
2
+
c
2
x
+
d
2
.
This gives
a
1
=
a
2
,
b
1
=
b
2
,
c
1
=
c
2
,
d
1
=
d
2
hence
~a
=
~
b
so
L
is onetoone.
Onto: For any
ax
3
+
bx
2
+
cx
+
d
∈
P
3
we have
L
±²
a b
c d
³´
=
ax
3
+
bx
2
+
cx
+
d
hence
L
is onto.
Thus,
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Vector Space

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