A [[Function]] $T: \R^m \to \R^n$ is a [[Linear Transformation]] if: - $\forall\vec x, \vec y \in \R^m$, $T(\vec x + \vec y) = T(\vec x) + T(\vec y)$ - $\forall \vec x, \vec y \in \R^m$, $T(k\vec x)=kT(\vec x)$ - $T(\vec 0)= \vec 0$ >[!info]- [[Geometry|Geometric]] Based Rules >- *Straight* [[Lines]] as *Straight* [[Lines]] $\in\Rn{2+}$ >- Planes as Planes $\in \Rn{3+}$ >- Preserve the parallel nature of vectors / lines / planes $\in \Rn n$ >- Parallelograms / Parallelepipeds >- Consistent “notching” on a line >- $\Rn2\mapsto\Rn2$ Will have *[[../../02 Areas/Math/Determinant|relative areas]]* >- $\Rn3\mapsto\Rn3$ Will have *[[../../02 Areas/Math/Determinant|relative volumes]]* >- $\vec 0 \mapsto \vec 0$ The [[Composition]] between two [[Linear Transformation]] is a [[Linear Transformation]]. >[!info] [[Proof]] >$\huge\begin{align*} >\let T&: \Rn m \mapsto \Rn n \\ >S &: \Rn n \mapsto \Rn n >\end{align*}$ >$\begin{align*} >(S\circ T)\pa{\vec x_{1} + \vec x_{2}} &= S\pa{T\pa{\vec x_{1}} + T\pa{\vec x_{2}}} \\ >&= (S\circ T)\pa{\vec x_{1}}+ (S \circ T) \pa{\vec x_{2}} \\ >(S \circ T) \pa{k\vec x} &= S\pa{kT\pa{\vec x}} \\ >&= k(S\circ T)\pa{\vec x} >\end{align*}$ Every [[Linear Transformation]] $T: \R^m \to \R^n$ has a *corresponding* [[Matrix]] $A \in M_{n\times m}$ such that $\forall \vec x \in \R^m:$ $T(\vec x)=A\vec x$. [[Universal Quantifier|For All]] [[Linear Transformation|Linear Transformations]] $T$, there is an unique [[Logical Inverse|Inverse]] $T^{-1}$ that is also a [[Linear Transformation]].