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).
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|
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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],](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FLinearTransformation%2FNumberedEquation1.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 a matrix multiplication only after specifying a vector
basis for 
and 
.
When 
and 
have an inner product, 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 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
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.
See also
Elliptic Fixed Point, General Linear Group, Hyperbolic Fixed Point, Invertible Linear Map, Involutory, Linear Algebra, Linear Operator, Matrix, Matrix Multiplication, Parabolic Fixed Point, Vector Basis, Vector Space Explore this topic in the MathWorld classroom
Portions of this entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd and Weisstein, Eric W. "Linear Transformation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LinearTransformation.html