Linear Transformation

Definition

Let:

$$\{\begin{array}{ll}A\in m\times n& \text{matrix}\\ \vec{x}\in {\mathbb{R}}^n& \text{input vector}\\ \vec{y}\in {\mathbb{R}}^m& \text{output vector}\end{array}$$

Thus, $A$ acts like a function, taking in any Vector in ${\mathbb{R}}^n$ and outputting a vector in ${\mathbb{R}}^m$.

This transformation is called a linear transformation if it maintains all properties of Vector Operations. That is:

1. $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$ for all $\vec{u},\vec{v}\in {\mathbb{R}}^n$
2. $T(c\vec{u})=cT(\vec{u})$ for all $\vec{u}\in {\mathbb{R}}^n$ and $c\in \mathbb{R}$

Where $T$ is the linear transformation defined by matrix $A$.