Transformation

A transformation or transform is a function ff that maps a set to itself. These functions are often seen in geometry and linear algebra or have geometric interpretations. For instance, a rotation is a transform. More generally, they can also be operators.

Properties

Since transformations are functions, they inherit all the properties of functions. A particularly important case is continuous transformations. An example of a continuous transformation φ\varphi is the rotation

(q1q2)→(cos⁡α−sin⁡αsin⁡αcos⁡α)(q1q2)=(φ1(q,α)φ2(q,α))\begin{pmatrix} q_{1} \\ q_{2} \end{pmatrix}\to \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix}\begin{pmatrix} q_{1} \\ q_{2} \end{pmatrix}=\begin{pmatrix} \varphi_{1}(q,\alpha) \\ \varphi_{2}(q,\alpha) \end{pmatrix}

where α∈[−π,π[\alpha \in[-\pi,\pi[. The derivative transformation ψ\psi is

(q˙1q˙2)→(cos⁡α−sin⁡αsin⁡αcos⁡α)(q˙1q˙2)=(ψ1(q,q˙,α)ψ2(q,q˙,α))\begin{pmatrix} \dot{q}_{1} \\ \dot{q}_{2} \end{pmatrix}\to\begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix}\begin{pmatrix} \dot{q}_{1} \\ \dot{q}_{2} \end{pmatrix}=\begin{pmatrix} \psi_{1}(q,\dot{q},\alpha) \\ \psi_{2}(q,\dot{q},\alpha) \end{pmatrix}

An example of a non-continuous transformation (called a discrete transformation) is

(q1q2)→(αq1αq2)\begin{pmatrix} q_{1} \\ q_{2} \end{pmatrix}\to \begin{pmatrix} \alpha q_{1} \\ \alpha q_{2} \end{pmatrix}

where α=±1\alpha=\pm 1.