Page 110 - Advanced Linear Algebra

94 Advanced Linear Algebra

The following theorem demonstrates one way in which the expression =°:
behaves like a fraction.

(
Theorem 3.8 The third isomorphism theorem ) Let be a vector space and
=
=
suppose that :‹ ; ‹ = are subspaces of . Then
=°: =
š
;°: ;
Proof. Let be defined by ²# b :³ ~ # b ; . We leave it to the
¢ = °: ¦ = °;
reader to show that is a well-defined surjective linear transformation whose

kernel is ;°: . The rest follows from the first isomorphism theorem.
The following theorem demonstrates one way in which the expression =°:
does not behave like a fraction.
Theorem 3.9 Let be a vector space and let be a subspace of . Suppose

:
=
=
. Then
that =~ = l = and : ~ : l : with : ‹ =
= = l = = =
~ š ^
: : l : : :

Proof. Let = ¦ ²=°:³ ^¢ =°:³ be defined by
²

²# b# ³ ~ ²# b: Á # b: ³

is direct. We leave it to
This map is well-defined, since the sum =~ = l =
the reader to show that is a surjective linear transformation, whose kernel is

:l : . The rest follows from the first isomorphism theorem.
Linear Functionals
(
Linear transformations from to the base field thought of as a vector space
-
=
)
over itself are extremely important.
Definition Let = be a vector space over - . A linear transformation
 ²= Á -³ whose values lie in the base field is called a linear functional
-
B
(or simply functional ) on . (Some authors use the term linear function ) . The
=
vector space of all linear functionals on is denoted by = * and is called the
=
algebraic dual space of .
=
The adjective algebraic is needed here, since there is another type of dual space
that is defined on general normed vector spaces, where continuity of linear
transformations makes sense. We will discuss the so-called continuous dual
space briefly in Chapter 13. However, until then, the term “dual space” will
refer to the algebraic dual space.