*eg.* Applying a [[Function|Transformation]] to a whole [[Lines|line]], [[Planes|plane]], *etc*, *etc*. $\huge \let T: \Rn n\to\Rn n$ #### Vector Form (Line or Plane) $\huge \begin{align*} l &: \vec x = P + t\vec v \\ T\pa{l} &= T\pa{P} + tT\pa{\vec v} \\ \end{align*}$ $\huge \begin{align*} \alpha &: \vec x = P + t\vec v + s\vec u \\ T\pa{\alpha} &= T\pa{P} + tT\pa{\vec v} + sT\pa{\vec u} \\ \end{align*}$ #### [[Normal Vector|Normal]] Form *You can’t use the same approach as before* because [[Linear Transformation]] preserve parallel lines but *not orthogonality*. >[!tip] A valid approach can be to convert normal form to vector form $\huge\begin{align*} l : n_{x}x + n_{y}y &= c \\ \vec n^{\intercal}\vec x &= c \\ \end{align*}$ To test if $\vec x$ is on $T(l)$: $\huge\begin{align*} T(l): \vec n ^{\intercal}\pa{\vec n^{\intercal} A^{-1}} \vec x &= c \\ \vec n ^{\intercal} A^{-1} &= \vec m ^{\intercal}\\ \pa{\vec n ^ {\intercal}A ^{-1}}^\intercal &= \vec m\\ \end{align*}$ >$\huge \vec m = \pa{A^{-1}}^{\intercal} \vec n $ >[!example] >$\huge\begin{align*} T&: \Rn3 \to \Rn3 \\ T&: \mat{x\\y\\z} \mapsto \mat{3x-y\\2y+z\\x+y+z} \\ \alpha &: x+2y-z=3\\ \end{align*}$ >$\huge\begin{align*} P &\in \alpha\\ \vec n &\perp \alpha\\ \vec v &\parallel \alpha\\ \vec u &\parallel \alpha\\ \vec n &= \mat{1\\2\\-1}\\ P &= \mat{3\\0\\0} \\ \vec v &= \mat{2\\-1\\0}\\ \vec u &= \mat{1\\0\\1}\\ \alpha: \mat{x\\y\\z} &= \mat{3\\0\\0} + t\mat{2\\-1\\0} + s\mat{1\\0\\1} \end{align*}$ >$\huge\begin{align*} \vec m &= T(\vec u) \times \vec T(\vec v) \\ &= \mat{-5\\-11\\13} \\ T: P &\mapsto \mat{9\\0\\3}\\ 6 &= 5(9) + 11(0) - 13(3) \\ T(\alpha)&: 5x+11y-13z = 6 \end{align*}$