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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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)
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then 
is a linear transformation from 
to 
, defined by
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(2)
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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
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(3)
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(4)
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Now rescale by defining 
and 
. Then the above equations become
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(5)
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where 
and 
,
,
,
and 
are defined in terms of the old constants. Solving for 
gives
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(6)
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so the transformation is one-to-one. To find the fixed points of the transformation, set 
to obtain
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(7)
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This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows.
