A linear map is a function from a vector space into a vector space that satisfies linearity. Specifically, for vectors and real numbers , we have . When , we call a linear operator.

A linear map is completely defined by its effect on the basis vectors. This is because the map is linear. Given an arbitrary vector , we have .

As an example, suppose that . Then

In the following program, we use the GUI to specify a linear operator. The text fields basis1, basis2, and basis3 indicate the vector each of the three basis vectors get mapped to. Each field should contain a sequence of three comma-separated numbers, such as 1,2,0. When you press Go, the linear operator is depicted by the volume into which the blue unit cube gets transformed, and also by the transformed red grid which can be toggled on or off. A sphere indicates the vector each basis vector maps to. For example, if the basis1 field contains the string 1,1,0 meaning , the ‘red’ basis vector maps to the vector indicated by the red sphere at position .

Linear operators

On the next page, we’ll use this program to visualize some linear operators.