>[!tip] Fun Fact: This is in [[Minecraft]] Parallel line $\vec n^\intercal \vec x =0$ with shear factor $\vec v$ $\huge I+ \frac{1}{\norms{n}}\vec v \vec n ^\intercal = I+ \vec v \hat n ^\intercal $ *or* $\huge I+ \frac{1}{\vec n^{\intercal} \vec P} \vector{PQ} \vec n^\intercal \iff \hat n \cdot \vector{PQ} = 0 $ >[!info] Symmetry >This transformation is *not* symmetrical #### Determinant $ \huge \det A = 1$ #### Inverse $\huge I - \frac{1}{\norms{n}}\vec v \vec n ^\intercal = I - \vec v \hat n ^\intercal $ $\huge I+ \frac{1}{\vec n^{\intercal} \vec Q} \vector{PQ} \vec n^{\intercal}\iff \hat n \cdot \vector{PQ} = 0 $ >[!example] >$\let T: \Rn2 \to \Rn2$ be the shear parallel to $l: 3x-y=0$ with shear factor $\mat{2\\6}$ > >$\huge\begin{align*} \vec n &= \mat{3\\-1}, \vec v = \mat{2\\6}\\ A&= I + \frac{1}{\sqrt{10}}\mat{2\\6}\mat{3&-1}\\ &= \frac{1}{\sqrt{10}}\mat{\sqrt{10} + 6 & -2 \\ 18 & \sqrt{10}- 6} \end{align*}$