### Linear
Transformation

##### Matrix of a linear transformation

Theorem Any linear transformation **T**: **R**^{n} → **R**^{m}
is also a matrix transformation, with the matrix given by

**A** = [**T**(**e**_{1})
**T**(**e**_{2}) ... **T**(**e**_{n})].

Proof By an earlier exercise,
any **x** ∈ **R**^{n} can be expressed as

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

Then by the linearity of **T**, we have

**T**(**x**)
= **T**(`x`_{1}**e**_{1}
+ `x`_{2}**e**_{2}
+ ... + `x`_{n}**e**_{n})
= `x`_{1}**T**(**e**_{1})
+ `x`_{2}**T**(**e**_{2})
+ ... + `x`_{n}**T**(**e**_{n}).

In another earlier exercise,
we already noticed the following equality

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

where **a**_{1} **a**_{2} ... **a**_{n}
are the columns of **A** and `x`_{1}, `x`_{2}, ..., `x`_{n}
are the coordinates of **x**. (You are recommended to try to verify the equality for
a general 3 by 3 matrix). Thus we conclude that

**T**(**x**)
= `Ax`
for **A** = [**T**(**e**_{1})
**T**(**e**_{2}) ... **T**(**e**_{n})].