# 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
• 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 #

### Rotation #

By $\varphi$ counter-clockwise

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

### Contraction/expansion #

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

• Horizontal contraction: $0 < a < 1$
• Horizontal expansion: $a > 1$
• Vertical contraction: $0 < b < 1$
• Vertical expansion: $b > 1$

### Shears #

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

• Horizontal shear: left $a < 0$, right $a > 0$
• Vertical shear: left $b < 0$, right $b > 0$