Def 1.3.2. A linear transformation $T: \R^n \to \R^m$ is a mapping such that for all scalars $a$ and all $\vec v, \vec w \in \R^n$,

$$ T(a\vec v) = aT(\vec v) \text{ and }T(\vec v + \vec w) = T(\vec v ) + T(\vec w) $$

Theorem 1.3.4.

1. Any $m\times n$ matrix $A$ defines a linear transformation $T: \R^n \to \R^n$ by matrix multiplication:

$$ T(\vec v) = A\vec v $$

2. Every linear transformation $T: \R^n \to \R^m$ is given by multiplication by the $m \times n$ matrix $[T]$:

$$ T(\vec v) = [T]\vec v $$

where the $i$th column of $[T]$ is $T(\vec e_i)$.

Proof that Matrix Multiplication is Linear

We prove that $A(B+C) =AB+BC$ and $(A+B)C = AC+BC$.

Suppose $A$ is a $n \times m$ matrix, and $B$ is a $m\times p$ matrix. We define matrix multiplication such that the $i$th $j$th element is given by $\sum_{k=1}^m a_{ik}b_{kj}$.

$$ (AB){ij} = \sum^m{k=1}a_{ik}b_{kj} $$

By this definition,

$$ (A(B+C)){ij} = \sum{k=1}^{m}a_{ij}(b_{kj}+c_{kj}) $$

$$ = (AB){ij}+(AC){ij} = \sum_{k=1}^m a_{ik}b_{kj} + \sum^m_{k=1}a_{ik}c_{kj} $$