### Composition and Matrix Multiplication

##### Extra: Composition of linear transformations is linear

Let **T**: **R**^{m} → **R**^{n}
and **S**: **R**^{k} → **R**^{n} be linear transformations.
Then by the definition, we have

**T**(**u** + **v**) =
**T**(**u**) + **T**(**v**),
**T**(`c`**u**) = `c`**T**(**u**),
**S**(**u** + **v**) =
**S**(**u**) + **S**(**v**),
**S**(`c`**u**) = `c`**S**(**u**).

To show the composition `TS`: **R**^{k} → **R**^{m}
is also linear means the verification of the following equalities:

**TS**(**u** + **v**) =
**TS**(**u**) + **TS**(**v**),

**TS**(`c`**u**) = `c`**TS**(**u**).

The first equality is verified as follows:

**TS**(**u** + **v**) =
**T**(**S**(**u** + **v**)) (definition of composition)

= **T**(**S**(**u**) + **S**(**v**)) (property 3. above)

= **T**(**S**(**u**)) + **T**(**S**(**v**)) (property 1. above)

= **TS**(**u**) + **TS**(**v**) (definition of composition)

The second equality is verified as follows:

**TS**(`c`**u**) =
**T**(**S**(`c`**u**)) (definition of composition)

= **T**(`Cs`(**u**)) (property 4. above)

= `c`**T**(**S**(**u**)) (property 2. above)

= `CTS`(**u**) (definition of composition)

This completes the proof that `TS` is a linear transformation.