## Linear Transformation

An Matrix is a linear transformation (linear Map) Iff, for every pair of -Vectors and and every Scalar ,

 (1)

and
 (2)

Consider the 2-D transformation

 (3) (4)

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

See also Elliptic Fixed Point (Map), Hyperbolic Fixed Point (Map), Involuntary, Linear Operator, Parabolic Fixed Point

References

Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 13-15, 1961.