LINEAR TRANSFORMATIONS
 Transformations
 A transformation T from R^{n} to R^{m}
takes
to
.
 Notation:
;
.
 R^{n} is the domain of T and R^{m} is the codomain.
 If
,
is the image of
using T.
 The set of all images is the range of T.
 Matrix Transformations
 If A is an
matrix, the transformation
has domain R^{n} and codomain R^{m}.
 Linear Transformations
 A transformation T is linear if:
 1.

for
in the domain of T.
 2.

for all
in the domain of T and scalars c.
 Matrices are linear transformations.
 Additional properties:
;
;
.
 Geometry applications:
 Shears, rotations, dilations, contractions
 The Matrix of a Linear Transformation:
 linear transformations can be represented by matrices:
if
and T is linear, then
with
 Onto and OneToOne

 Defintion: a transformation
is onto R^{m}
if, for every
,
there exists at least one
with
.
 Defintion: a transformation
is onetoone
if, for each
,
there is at most one
with
.
 Theorem: a transformation
is onetoone iff
the equation
has only the trivial solution.
 Theorem: if
with standard matrix A:
 T is onto iff the columns of A span R^{m}.
 T is onetoone iff the columns of A are linearly independent.
Alan C Genz
19990916