Page 111 - Advanced Linear Algebra
P. 111
The Isomorphism Theorems 95
To help distinguish linear functionals from other types of linear transformations,
we will usually denote linear functionals by lowercase italic letters, such as ,
and .
Example 3.1 The map ¢ -´%µ ¦ - defined by ² ²%³³ ~ ² ³ is a linear
functional, known as evaluation at .
Example 3.2 Let 9´ Á µ denote the vector space of all continuous functions on
9. Let
´ Á µ s ¢ ´ Á µ ¦ s be defined by
² ²%³³ ~ ²%³ %
9
Then ´ Á µ i .
For any = * , the rank plus nullity theorem is
²
dim ker² ³³ b dim²im ² ³³ ~ dim²= ³
But since im² ³ - , we have either im² ³ ~ ¸ ¹ , in which case is the zero
linear functional, or im² ³ ~ - , in which case is surjective. In other words, a
nonzero linear functional is surjective. Moreover, if £ , then
=
codim ² ²ker ³ ³ ~ dim 6 ~ 7
ker ² ³
and if dim²= ³ B , then
dim ker ² ³ ³ ~ dim = ² ³ c
²
Thus, in dimensional terms, the kernel of a linear functional is a very “large”
subspace of the domain .
=
The following theorem will prove very useful.
Theorem 3.10
)
1 For any nonzero vector #= , there exists a linear functional = * for
which ²#³ £ .
)
2 A vector #= is zero if and only if ²#³~ for all = * .
)
3 Let = i . If ²%³ £ , then
=~ º%» l ker ² ³
)
4 Two nonzero linear functionals Á = i have the same kernel if and only
if there is a nonzero scalar such that ~ .
Proof. For part 3 , if £ # º%» q ker ² ³ , then ²#³ ~ and # ~ % for
)
£ -, whence ²%³ ~ , which is false. Hence, º%» q ker ² ³ ~ ¸ ¹ and
the direct sum :~ º%» l ker ² ³ exists. Also, for any # = we have
