Definition

You define any transform in 2 dimensions with the vector and it describes how a point is changed. The new point is called an image. A linear transform is a Vector transform with only linear terms and no constants.

They have the properties that:

  • They can be represented by a Matrix

A-Level Further Core 1

Reflections and Rotations

You can describe any linear transform just by the effect it has on unit vectors as every vector is a linear combination of the unit vectors. Points/lines that don’t change under the transform are called invariant. The matrix maps and The matrix of a rotation through angle anticlockwise about the origin

Enlargement and Stretches

You can represent a stretch with matrix It has stretch factor parallel to the -axis and stretch factor parallel to the -axis. For stretches only along the -axis, points on the -axis are invariant and the line is invariant and vice versa. For stretches in both direction the only invariance is the origin For a linear transform by matrix , is the scale factor of area (if it’s negative the shape has been reflected)

Reflections

Rotations

Successive transformations

The matrix represents the singular transform of the result of a transform by then

Linear Transforms in 3D

For a linear transform by a given 3x3 Matrix

Inverting Transforms

Since , describes the inverse transformation of