Let V and W be vector spaces. Recall that a function T:V® W is called a linear transformation if it preserves both vector addition and scalar multiplication:

If V = R² and W = R², then T:R² ® R² is a linear transformation if and only if there exists a 2 ×2 matrix A such that T(v) = Av for all v Î R². Matrix A is called the standard matrix for T. The columns of A are T [ 1 0 ]^T and T [ 0 1 ]^T, respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parallelograms to parallelograms. We will often illustrate the action of a linear transformation T:R²® R² by looking at the image of a unit square under T.

Rotations

The standard matrix for the linear transformation T:R² ®R² that rotates vectors by an angle q is

Reflections

For every line in the plane, there is a linear transformation that reflects vectors about that line. Reflection about the x-axis is given by the standard matrix

Expansions and Compressions

Shears

The standard matrix

Similarly,

• If finitely many linear transformations from R² to R² are performed in succession, then there exists a single linear transformation with the same effect.

• If the standard matrix for a a transformation T: R²® R² is invertible, then it can be shown that the geometric effect of T is the same as some sequence of reflections, expansions, compressions, and sheers.

For every linear transformation T: R² ® R² of the plane, there exists a standard matrix A such that