Transformations

A transformation/function/mapping is a rule that assigns to each vector $\vec{x}$ in $\mathbb{R}^n$ a vector $T(\vec{x})$ in $\mathbb{R}^m$

$$ T : \mathbb{R}^n \rightarrow \mathbb{R}^m $$

Domain of $T$: $\mathbb{R}^n$
Codomain of $T$: $\mathbb{R}^m$
Image of $\vec{x}$: $T(\vec{x}) \in \mathbb{R}^m$
Range: set of all images $T(\vec{x})$
Linear transformation: has a unique $m \times n$ standard matrix $A$ such that $T(\vec{x}) = A\vec{x}$ for all $\vec{x}$ in $\mathbb{R}^n$
- Every $j$th column is the vector $T(e_j)$ where $j$ is the $j$th column of the identity matrix: $$A = [~~T(\vec{e_1})~~\dots~T(\vec{e_n})~]$$
- Use this property to find $A$ given $\vec{v}$ and $T(\vec{v})$: $$\vec{v} = ae_1 + be_2 \Leftrightarrow T(\vec{v}) = aT(e_1) + bT(e_2)$$
One-to-one (injective): if each $\vec{b}$ in $\mathbb{R}^m$ is the image of at most one $\vec{x}$ in $\mathbb{R}^n$ (every vector is mapped to a different output)
- $T$ is one-to-one if and only if $T(\vec{x})=\vec{0}$ has only the trivial solution
- $T$ is one-to-one if and only if the columns of the standard matrix are linearly independent
Onto (surjective): if each $\vec{b}$ in $\mathbb{R}^m$ is the image of at least one $\vec{x}$ in $\mathbb{R}^n$ (all possible outputs can be reached)
- $T$ is onto if and only if the columns of the standard matrix span $\mathbb{R}^m$

Geometric transformations #

By $\varphi$ counter-clockwise

$$ \begin{pmatrix} \cos{\varphi} & -\sin{\varphi} \\\ \sin{\varphi} & \cos{\varphi} \end{pmatrix} $$

$$ \begin{pmatrix} a & 0 \\\ 0 & b \end{pmatrix} $$

$$ \begin{pmatrix} 1 & a \\\ b & 1 \end{pmatrix} $$