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Text 1. A linear transformation between two vector spaces and is a map such that the following hold:
|
|
A linear transformation between two vector spaces 
and 
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 
such that 
. It is always the case that 
. Also, a linear transformation always maps lines to lines (or to zero).
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
| (1) |
then is a linear transformation from 
to 
, defined by
| (2) |
When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector space basis for and . When and have an inner product, and their vector space 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 space of polynomials in one variable, and be the derivative. Then 
, which is not continuous because 
while 
does not converge.
Linear two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation
| 
| 
| (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) |
This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows.
| variables | type |
| hyperbolic fixed point |
| elliptic fixed point |
| parabolic fixed point |
*Source: Rowland, Todd and Weisstein, Eric W. " Linear Transformation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LinearTransformation.html