Page 75 - Advanced Linear Algebra

Chapter 2

Linear Transformations

Linear Transformations

Loosely speaking, a linear transformation is a function from one vector space to
another that preserves the vector space operations. Let us be more precise.

Definition Let and > be vector spaces over a field . A function ¢ = ¦ >
=
-
is a linear transformation if

² " b #³ ~ ²"³ b ²#³
for all scalars Á  - and vectors " ,#  = . The set of all linear
transformations from to > is denoted by ² B = Á > . ³
=
1 A linear transformation from to is called a linear operator on . The
)
=
=
=
set of all linear operators on is denoted by ² B = ³ . A linear operator on a
=
real vector space is called a real operator and a linear operator on a
complex vector space is called a complex operator .
(
)
2 A linear transformation from to the base field thought of as a vector
=
-
space over itself is called a linear functional on . The set of all linear
)
=
functionals on is denoted by = i and called the dual space of .
=
=
We should mention that some authors use the term linear operator for any linear
transformation from to > . Also, the application of a linear transformation
=
#
on a vector is denoted by ² # ³ or by # , parentheses being used when

necessary, as in , or to improve readability, as in ²" b #³ ² "³ rather than
²²"³³.
Definition The following terms are also employed:
)
1 homomorphism for linear transformation
)
2 endomorphism for linear operator
)
3 monomorphism or embedding ) for injective linear transformation
(
4 epimorphism for surjective linear transformation
)
)
5 isomorphism for bijective linear transformation.