Linear Transformation

A linear transformation between two vector spaces V and W is a map T:V->W such that the following hold:

1. T(v_1+v_2)=T(v_1)+T(v_2) for any vectors v_1 and v_2 in V, and

2. T(alphav)=alphaT(v) for any scalar alpha.

A linear transformation may or may not be injective or surjective. When V and W have the same dimension, it is possible for T to be invertible, meaning there exists a T^(-1) such that TT^(-1)=I. It is always the case that T(0)=0. 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 n×m matrix A, define T(v)=Av, where v is written as a column vector (with m coordinates). For example, consider

 A=[0 1; -2 2; 1 0],

then T is a linear transformation from R^2 to R^3, defined by


When V and W are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector basis for V and W. When V and W have an inner product, and their vector bases, {v_1,...,v_m} and {w_,...,w_n}, are orthonormal, it is easy to write the corresponding matrix A=(a_(ij)). In particular, a_(ij)=<w_i,T(v_j)>. Note that when using the standard basis for R^n and R^m, the jth column corresponds to the image of the jth standard basis vector.

When V and W are infinite dimensional, then it is possible for a linear transformation to not be continuous. For example, let V be the space of polynomials in one variable, and T be the derivative. Then T(x^n)=nx^(n-1), which is not continuous because x^n/n->0 while T(x^n/n) does not converge.

Linear two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation


Now rescale by defining lambda=x_1/x_2 and lambda^'=x_1^'/x_2^'. Then the above equations become


where alphadelta-betagamma!=0 and alpha, beta, gamma, and delta are defined in terms of the old constants. Solving for lambda gives


so the transformation is one-to-one. To find the fixed points of the transformation, set lambda=lambda^' to obtain


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 Web Resource.

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