A linear transformation between two vector spaces 
and 
over the same field is a map 
such
that the following hold:
1. 
for any vectors 
and 
in 
, and
2. 
for any scalar 
.
A linear transformation may or may not be injective or surjective. When 
and 
have the same dimension, it
is possible for 
to be invertible, meaning there exists a map 
such
that 
and 
,
where 
and 
are the identity maps on 
and 
, respectively. It is always the case that 
. Also, a linear transformation always maps lines to lines
or to single points.
The main example of a linear transformation is given by matrix multiplication. Given an 
matrix 
, define 
, where 
is written as a column vector
(with 
coordinates). For example, consider
![A=[0 1; -2 2; 1 0],](/images/equations/LinearTransformation/NumberedEquation1.svg) |
(1)
|
then 
is a linear transformation from 
to 
, defined by
 |
(2)
|
When 
and 
are finite-dimensional,
a general linear transformation can be written as matrix
multiplication only after specifying a vector basis
for 
and 
.
When 
and 
have an inner product linear in the second argument,
and their vector bases, 
and 
, are orthonormal,
it is easy to write the corresponding matrix 
. In particular, 
. Note that when using the standard
basis for 
and 
,
the 
th
column corresponds to the image of the 
th standard basis vector.
When 
and 
are infinite-dimensional,
then it is possible for a linear transformation to not be continuous.
For example, let 
be the vector space of real polynomials
on ![[0,1]](/images/equations/LinearTransformation/Inline46.svg)
with the supremum norm 
, and let 
be differentiation. Then
.
Since 
while 
for 
,
the sequence 
converges to zero but its images
do not converge to zero. Thus 
is not continuous in this norm.
Linear two-dimensional transformations have a simple classification. Consider an invertible
two-dimensional linear transformation with 
 |  |  |
(3)
|
 |  |  |
(4)
|
Now rescale by defining 
and 
. Then the above equations
become
 |
(5)
|
where 
and 
,
,
,
and 
are defined in terms of the old constants. Solving for
gives
 |
(6)
|
so the transformation is one-to-one. To find the fixed points of the transformation, set 
to obtain
 |
(7)
|
For a nonidentity transformation on the complex projective line, this gives two fixed points, counted with
multiplicity and including the point
at infinity when needed. The identity fixes every
point. For real coefficients
with 
,
the nonidentity linear fractional
transformations are classified as follows.
