If $L:\mathbb{R}^3\to\mathbb{R}^3$, be the reflection with respect to the plane spanned by $\left\{\begin{bmatrix} -2\\-7\\ -8\end{bmatrix},\begin{bmatrix} 10\\ 2\\ 7\end{bmatrix}\right\}$. Find the matrix of L. Hint: First find the matrix $[L]_{\mathbb{B}}^{\mathbb{B}}$ in a basis that contains the two given vectors and their crossed product. Then use transition matrices.
I googled how to do do use reflection matrices across a plane, but I couldn't follow why anything they were saying worked. Thanks for any help.
Best Answer
Hint: Let $e,f$ be the given two vectors spanning the plane, and set $g=e\times f$ which is orthogonal to the plane.
As the hint says, $\mathcal B:=(e, f, g)$ is a basis for $\Bbb R^3$.
By the geometric properties of the reflection, we know that $$L(e)=e, \ \ L(f)=f, \ \ L(g)=-g$$ So the matrix of $L$ coordinated in basis $\mathcal B$ is $$[L]_{\mathcal B}=\pmatrix{1&0&0\\0&1&0\\0&0&-1}$$ since this contains the $\mathcal B$-coordinates of $L(e),\, L(f),\ L(g)$ in the columns.
Then apply basis transformation: $$[L]=B\, [L]_{\mathcal B}\, B^{-1}$$ where $B$ is the matrix with columns $e,f,g$.