Linear Transformations

Introduced to in Math 115

• Linear Transformation - If for every $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and for every $c\in \mathbb{R}$
  • 1. $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$
  • 2. $T(c\vec{x})=cT(\vec{x})$

[!Linearity Test]

A function $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation if and only if

$T(c_1\vec{x}+c_2\vec{y})=c_{1}T(\vec{x})+c_{2}T(\vec{y})$

for all $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and for all $c_1,c_2\in \mathbb{R}$

[!Properties of Matrix Transformations]

Let $A\in M_{mxn}(\mathbb{R})$ and let $f_A$ be the matrix transformation corresponding to A. For every $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and for every $c\in \mathbb{R}$

• $f_A(\vec{x}+\vec{y})=f_A(\vec{x})+f_A(\vec{y})$
• $f_A(c\vec{x})=c{f}_A(\vec{x})$

[!Definition 5.1.4]

A function $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is called a linear transformation (or a linear mapping) if for every $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and for every $c\in \mathbb{R}$, we have

• $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$
• $T(c\vec{x})=cT(\vec{x})$

[!Theorem 5.1.10]

If $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation, then T is a matrix transformation with corresponding matrix

$[T]=[T({\vec{e}}_1)\dots T({\vec{e}}_n)]\in M_{mxn}(\mathbb{R})$

$T(\vec{x})=[T]\vec{x}$ for every $\vec{x}\in {\mathbb{R}}^n$

This is also known as the standard matrix