### Linear
Transformation

##### Matrix transformations are linear

Theorem **A**(**x** + **y**) = **Ax** + **Ay**,
**A**(`c`**x**) = `c`**Ax**.

Proof By an earlier exercise,
we have

`x`_{1}**a**_{1} + `x`_{2}**a**_{2}
+ ... + **x**_{n}**a**_{n} = `Ax`

where **A** = [**a**_{1} **a**_{2}
... **a**_{n}] is a matrix with the vectors **a**_{1},
**a**_{2}, ..., **a**_{n} as columns, and `x`_{1},
`x`_{2}, ..., `x`_{n} are the coordinates of **x**. Thus

**Ax** + **Ay**
= `x`_{1}**a**_{1}
+ `x`_{2}**a**_{2}
+ ... + **x**_{n}**a**_{n}
+ `y`_{1}**a**_{1}
+ `y`_{2}**a**_{2}
+ ... + **y**_{n}**a**_{n}

= (`x`_{1}**a**_{1}
+ `y`_{1}**a**_{1})
+ (`x`_{2}**a**_{2}
+ `y`_{2}**a**_{2})
+ ... + (**x**_{n}**a**_{n}
+ **y**_{n}**a**_{n})

= (`x`_{1} + `y`_{1})**a**_{1}
+ (`x`_{2} + `y`_{2})**a**_{2}
+ ... + (**x**_{n} + **y**_{n})**a**_{n}

Since `x`_{1} + `y`_{1}, `x`_{2} + `y`_{2}, ...,
`x`_{n} + `y`_{n} are the coordinates of **x** + **y**,
we see that the summation is **A**(**x** + **y**).

The proof for **A**(`c`**x**) = `c`**Ax** is similar.