# Content:Combinations of Linear Transformations

A linear transformation moves points from an xy-grid onto new locations on the xy-grid, according to a transformation matrix M.

We know certain basic types of transformation that are relatively easy to understand and work with. For example: enlargement, reflection, rotation, shear.

We can combine these to build a composite transformation by applying a sequence of these basic transformations.

## Prerequisites

We'll assume you already know about:

## Order Matters!

First of all, it's important to note that the order of a sequence of transformations matters - like matrix multiplication, it is NOT commutative. If you change the order of transformations, you will typically get different results.

If we do a shear followed by a reflection, that will give different results than a reflection followed by a shear.

## Sequence of two transformations

Suppose we have two transformation matrices and , and a point with coordinates , which we'll represent in position vector form: .

We want to apply the two transformation matrices one-by-one: first A, then B.

To apply transformation A, we need to pre-multiply by the matrix A, so .

To then apply a second transformation matrix , this also has to be pre-multiplied. will go in first place: .

Note that matrix multiplication is **not commutative** - so we cannot swap the order of and . must come before .

However, matrix multiplication **is associative** - we can process the multiplications in two different orders:

- first calculate and then pre-multiply by B, giving , or,
- first calculate and then pre-multiply by , also giving .

### Two Transformations in One

The transformation followed by the transformation can be represented by the transformation (**note the order!**).

We could now think about this as the single *composite* transformation .

### Interpreting Composite Transformations: Order Matters

When we have a composite transformation, such as , we may want to think about it also as a sequence. The transformation closest to the position vector always happens first in the sequence.

When we apply the composite transformation, we have to pre-multiply by , giving: .

Thinking about the decomposed transformations and , we see is closest to the position vector, so we know transformation happens first. Then, transformation happens second.

## Next Steps

- Lines of Invariant Points, Invariant Lines
- Types of Linear Transformation
- Combinations of Linear Transformations
- Area and Volume in Linear Transformations