Page 76 - Advanced Linear Algebra

60 Advanced Linear Algebra

)
6 automorphism for bijective linear operator.
Example 2.1
)
=
1 The derivative +¢ = ¦ = is a linear operator on the vector space of all
infinitely differentiable functions on .
s
)
2 The integral operator ¢-´%µ ¦ -´%µ defined by
%
~ ²!³ !

is a linear operator on -´%µ .
3 Let be an d matrix over . The function ( ¢ - ¦ - defined by
)
-
(
( #~(#, where all vectors are written as column vectors, is a linear
transformation from - to - . This function is just multiplication by .
(
)

4 The coordinate map ¢= ¦ - of an -dimensional vector space is a
linear transformation from to - .
=
The set B is a vector space in its own right and ²= ³ has the structure of
B²= Á > ³
an algebra, as defined in Chapter 0.
Theorem 2.1
)
1 The set B²= Á > ³ is a vector space under ordinary addition of functions
and scalar multiplication of functions by elements of .
-
)
B

2 If ²<Á = ³ and ²= Á > ³ , then the composition is in ²<Á > ³ .
B
B
)
B

3 If ²= Á > ³ is bijective then c B ²> Á = . ³
)
4 The vector space B²= ³ is an algebra, where multiplication is composition
of functions. The identity map B ²= ³ is the multiplicative identity and
the zero map  ²= ³ is the additive identity.
B
)
Proof. We prove only part 3 . Let ¢= ¦ > be a bijective linear
transformation. Then c ¢> ¦ = is a well-defined function and since any two
vectors $ and $ in > have the form $ ~ # and $ ~ # , we have
c c ² $ b $ ³ ~ ² # b # ³

~ c ² ² # b # ³³

~ # b #
c ²$ ~ c ²$ ³
³ b
which shows that c is linear.
One of the easiest ways to define a linear transformation is to give its values on
a basis. The following theorem says that we may assign these values arbitrarily
and obtain a unique linear transformation by linear extension to the entire
domain.

Theorem 2.2 Let = and > be vector spaces and let ~ ¸ #8 “  0 ¹ be a
basis for = . Then we can define a linear transformation  B ² = Á > ³ by