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.
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