Page 78 - Advanced Linear Algebra
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62 Advanced Linear Algebra
Isomorphisms
Definition A bijective linear transformation ¢= ¦ > is called an
isomorphism from to > . When an isomorphism from to > exists, we say
=
=
that and > are isomorphic and write = > .
=
8
Example 2.2 Let dim²= ³ ~ . For any ordered basis of , the coordinate
=
map 8 ¢= ¦ - that sends each vector # = to its coordinate matrix
´#µ - is an isomorphism. Hence, any -dimensional vector space over is
-
8
isomorphic to - .
Isomorphic vector spaces share many properties, as the next theorem shows. If
B ²= Á > ³ and : = we write
:~ ¸ :¹
Theorem 2.4 Let ²= Á > ³ be an isomorphism. Let : = . Then
B
1 : ) spans if and only if : spans > .
=
2 : ) is linearly independent in if and only if : = is linearly independent in
> .
=
3 : ) is a basis for if and only if : is a basis for > .
An isomorphism can be characterized as a linear transformation ¢= ¦ > that
maps a basis for to a basis for > .
=
Theorem 2.5 A linear transformation B ²= Á > ³ is an isomorphism if and
only if there is a basis for for which 8 = is a basis for > . In this case,
8
maps any basis of to a basis of > .
=
The following theorem says that, up to isomorphism, there is only one vector
space of any given dimension over a given field.
Theorem 2.6 Let and > be vector spaces over . Then = > if and only
-
=
if dim²= ³ ~ dim²> . ³
In Example 2.2, we saw that any -dimensional vector space is isomorphic to
- ) . Now suppose that is a set of cardinality and let ² - ) ³ be the vector
space of all functions from to with finite support. We leave it to the reader
-
)
) by
to show that the functions ²- ³ defined for all )
% ~ if
²%³ ~ F
% £ if
) , called the standard basis . Hence, dim ) .
((
form a basis for ²- ³ ²²- ³ ³ ~ )
It follows that for any cardinal number , there is a vector space of dimension .
)
Also, any vector space of dimension is isomorphic to ²- ³ .
