# Linear Transformation

This has extremely important applications for Robotics and Virtual Reality. Used for Rigid-Body Transformations

Notes from MATH115, Articulated Robotics, and Steven M. Lavalleβs VR lectures.

Think of linear transformations as just functions in higher dimensions, i.e. $R_{n}$. We often represent them as a Matrix Transformation. For example, a 2D Linear Transformation: $[x_{β²}y_{β²}β]=[12β12β][xyβ]$

### Linear Transformation

A function $L:R_{n}βR_{m}$ is called a linear transformation if for every $x$,$yββR_{n}$ and for every $s,tβR$ we have $L(sx+tyβ)=sL(x)+tL(yβ)$

Now, whatβs super cool is that we can represent a linear transformation as a Matrix Transformation!

If $L:R_{n}βR_{m}$ is a linear transformation, then $L$ is a Matrix Transformation with corresponding matrix $[L]$, i.e. $L(x)=[L]x$ for every $xβR_{n}$. The matrix $[L]$ is given by $[L]=[L(e_{1})β―L(e_{n})]βM_{mΓn}(R)$

Proof is at page 201 of the pdf.

##### Some Properties

- 0 always maps to 0, i.e. $L(0)=0$
- Linear Transformations are always an Odd Function, i.e. $L(βx)=βL(x)$
- You can do Composition of Function on Linear Transformation, where $M(L(x))=[MβL](x)=[M][L](x)$

- This is super useful, for example, when you apply multiple rotations, see Rotation Matrix

### Inverse Linear Transformations

This is actually more important than I thought. For example, if you apply a rotation linear transformation, how do you undo that rotation?

- Answer: Represent linear transformation as Matrix Transformation, and get the inverse of the matrix. Proof at page 215

if $L:R_{n}βR_{n}$ is an invertible linear transformation, then $[L_{β1}]=[L]_{β1}$

### Kernel and Range of Linear Transformations

Given a linear transformation $L:R_{n}βR_{m}$, there are two important sets that carry with them information about $L$.

**Kernel Definition**: Let $L:R_{n}βR_{m}$ be a (linear) transformation. The kernel of $L$ is
$Ker(L)={xβR_{n}β£L(x)=0}$
Note that $Ker(L)βR_{n}$. It is a subspace of it.

**Range Definition**: Let $L:R_{n}βR_{m}$ be a (linear) transformation. The range of $L$ is
$Range(L)={L(x)β£xβR_{n}}$

Note that $Range(L)βR_{m}$. It is a subspace of it.

Theorem 35.6. Let $L:R_{n}βR_{m}$ be a linear transformation with standard matrix $[L]$. Then $Ker(L)=Null([L])$ $Range(L)=Col([L])$

### One-to-One and Onto Linear Transformations

Wow, as I revisit these notes, This is Serendipity with the stuff I learned in SE212 about what Functions are.

If $L:R_{n}βR_{m}$ is a linear transformation, then $L$ is one-to-one if and only if $Ker(L)={0}$.

Let $L:R_{n}βR_{m}$ be a linear transformation. Then L is one-to-one if and only if $rank([L])=n$.

$L$ is a one-to-one correspondence if and only if $L$ is invertible.