# Lecture 8: Coordinate Transformations

## Basics

Multiplying a matrix by a vector allows us to produce a new, changed vectors.

Such a transformation can be used to **scale**, **shear**, and **rotate** a vector or Cartesian coordinates.

## Scale

### Matrix formulation:

## Shear

### Matrix formulation:

## Rotation

### Matrix formulation:

### Matrix derivation:

There is a requisite diagram for understanding this which is coming soonâ€¦

is rotated by

makes angle with the x-axis and its length is

Therefore:

And:

Using trigonometric relationships:

Thus:

Substituting the equations for and :

And the transformation matrix is revealed.