Page 79 - Advanced Linear Algebra

Linear Transformations 63

Theorem 2.7 If is a natural number, then any -dimensional vector space

)
-

over is isomorphic to - . If is any cardinal number and if is a set of
cardinality , then any -dimensional vector space over is isomorphic to the

-

)
-
)
vector space ²- ³ of all functions from to with finite support.
The Rank Plus Nullity Theorem
Let ²= Á > ³ . Since any subspace of has a complement, we can write

B
=
=~ ker ² ³ l ker ² ³

where ker²³ is a complement of ker²³ in . It follows that
=

dim²= ³ ~ dim ker² ³³ b dim ker² ³ ³

²
²

Now, the restriction of to ker²³ ,
² ker ¢ ³ ¦ >
is injective, since

ker² ³ ~ ker² ³ q ker² ³ ~ ¸ ¹

Also, im²³ ‹ im² ³ . For the reverse inclusion, if #  im² ³ , then since

# ~ " b $ " for  ² ker ³ $ and  ² ker ³ , we have

im
#~ " b $~ $~ $ ² ³

Thus im²³ ~ im² ³ . It follows that

ker²³ š im ²³

From this, we deduce the following theorem.
Theorem 2.8 Let B ²= Á > . ³
)

1 Any complement of ker²³ is isomorphic to im²³
2)(The rank plus nullity theorem )
dim ker ³ ² ² ³ b dim im ² ³ ³ ~ dim = ² ³
²
or, in other notation,

rk²³ b null²³ ~ dim ²= ³
Theorem 2.8 has an important corollary.
Corollary 2.9 Let B  ²= Á > ³ , where dim ²= ³~ dim ²> ³B . Then is

injective if and only if it is surjective.
Note that this result fails if the vector spaces are not finite-dimensional. The
reader is encouraged to find an example to support this statement.