Preserves everything that [[Linear Transformation]] preserve with the exception of *the origin point can move in an Affine transformation*. > ![[Pasted image 20231102111415.png]] $\let T: \Rn n \to \Rn n$, be an affine transformation. There is a coressponding $n\times n$ [[Matrix]] A, and a “translation [[Vector]]” $\vec b \in \Rn n$ such that for any [[Point]] $\vec x$: $\huge T(\vec x) = \underbrace{A\vec{\mathbf{x}}}_\text{Linear Transformation} + \underbrace{ \vec b }_\text{Translation}$ ## [[Composition]] $\huge\begin{align*} T_1&: \vec{\mathbf{x}} \mapsto A_{1}\vec{\mathbf{x}} + \vec b_{1}\\ T_2&: \vec{\mathbf{x}} \mapsto A_{2}\vec{\mathbf{x}} + \vec b_{2}\\ \end{align*}$ $\huge\begin{align*} T_{2} \circ T_{1} &= T_{2}\pa{ A_{1} \vec{\bf{x}} + \vec b _{1} } \\ &= T_{2}(A_{1}\vec{\bf x}) + T_{2}(\vec b_{1}) \\ &= A_{2}A_{1} \vec{\bf x} + \vec b_{2} + A_{2} \vec b_{1} + \vec b_{2} \\ &= A_{2}A_{1} \vec{\bf x} + A_{2} \vec b_{1} + 2\vec b_{2} \\ &= A_{2}\pa{A_{1} \vec{\bf x} + \vec b_{1} }+ 2\vec b_{2} \end{align*}$