Free Professional Development

Or

Author:
c o

To introduce the concept of a linear transformation

To understand the properties of linear transformations

To explore linear transformations by applying them to figures in two dimensional space

The learner is introduced to the concept of a linear transformation, which is then applied to transforming shapes in the plane.

Tutorial

Before diving into this packet, the learner should be familiar with the concept of a function, with matrix multiplication, and should be comfortable with the basic notion of vectors and vector spaces.

The following examples show that multiplication by a matrix really is a linear transformation. First, we show that multiplication by a constant can occur before or after applying the matrix:

Next we see that the map of the sum is the sum of the maps:

We can think of *2x2* matrices as transformations from * R^{2}* to

__ Scaling__ can occur horizontally, vertically, or both. The basic horizontal stretch is given by a matrix like this: , which moves a point

__ Reflection__ moves a point from one side of a line to the other, and places it at a distance equal to the distance from the line to the original point.. A basic reflection across the x-axis is given by the matrix , which only changes the sign of the y-coordinate. Likewise, reflection across the y-axis is given by the matrix , which changes the sign of the x-coordinate.

__ Rotation__ moves a point circularly about the origin. The matrix rotates the plain counter-clockwise by an angle of

Another way to alter points in a plane is to add a vector to them. That is, if we are given the point * (1,2)*, and we have a vector