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# 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).

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 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

 (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.

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 Web Resource. https://mathworld.wolfram.com/LinearTransformation.html