I was working on an exercise in which I am given a vector $(2,-1, 2)$ and I am supposed to find the standard matrix $A$ of the reflection operator $T$ on $\mathbb{R}^3$ such that $T(v)=-v$.
Here's my attempt at the problem:
$$2x_1-x_2+2x_3=0$$
I use it to get $(1,2,0)$ and $(0,-2,1)$ (the two columns of the new matrix $C$) as part of the basis. After that however; I'm confused what I should do. Does anyone have any ideas? I'm not clear so any tips directing me in the right direction would be much appreciated. Thanks.
Best Answer
Surely the matrix $A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ (or in index notation, $A_{ij} = -\delta_{ij}$) will transform any vector $\mathbf{v}$ to $-\mathbf{v}$, since it will reverse all its components.