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. Lavalle’s VR lectures.

Think of linear transformations as just functions in higher dimensions, i.e. ${\mathbb{R}}^n$. We often represent them as a Matrix Transformation. For example, a 2D Linear Transformation: $\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

Linear Transformation

A function $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is called a linear transformation if for every $\vec{x}$,$\vec{y}\in {\mathbb{R}}^n$ and for every $s,t\in \mathbb{R}$ we have $L(s\vec{x}+t\vec{y})=sL(\vec{x})+tL(\vec{y})$

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

If $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation, then $L$ is a Matrix Transformation with corresponding matrix $[L]$, i.e. $L(\vec{x})=[L]\vec{x}$ for every $\vec{x}\in {\mathbb{R}}^n$. The matrix $[L]$ is given by $[L]=[L({\vec{e}}_1)\cdots L({\vec{e}}_n)]\in M_{m\times n}(\mathbb{R})$

Proof is at page 201 of the pdf.

Some Properties

1. 0 always maps to 0, i.e. $L(\vec{0})=\vec{0}$
2. Linear Transformations are always an Odd Function, i.e. $L(-\vec{x})=-L(\vec{x})$
3. You can do Composition of Function on Linear Transformation, where $M(L(\vec{x}))=[M\circ L](\vec{x})=[M][L](\vec{x})$

• 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 $L:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is an invertible linear transformation, then $[L^{-1}]=[L{]}^{-1}$

Kernel and Range of Linear Transformations

Given a linear transformation $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$, there are two important sets that carry with them information about $L$.

Kernel Definition: Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a (linear) transformation. The kernel of $L$ is $Ker(L)=\{\vec{x}\in {\mathbb{R}}^n\mid L(\vec{x})=\vec{0}\}$ Note that $Ker(L)\subseteq {\mathbb{R}}^n$. It is a subspace of it.

Range Definition: Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a (linear) transformation. The range of $L$ is $Range(L)=\{L(\vec{x})\mid \vec{x}\in {\mathbb{R}}^n\}$

Note that $Range(L)\subseteq {\mathbb{R}}^m$. It is a subspace of it.

Theorem 35.6. Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ 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 $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation, then $L$ is one-to-one if and only if $Ker(L)=\{\vec{0}\}$.

Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ 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.