Linear Transformation
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
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(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 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
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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.
| variables | type |
| | hyperbolic fixed point |
| | elliptic fixed point |
| | parabolic fixed point |