Find the matrices $S_1$ and $S_2$ of the reflections in $\mathbb{R}^3$ according to the following planes defined by the following subspaces: $V_1=\{\vec{x}:x_1+x_2-x_3=0\}$ \ $ V_2=\{\vec{x}:x_1-x_2+x_3\}$
And find the properties of the transformation $T=S_1S_2$
At first it had no sense since it seemed a very general problem, but now that I think about it, does it has to be done with the formula: $$R=\cos\theta\mathbf{1}+(1-\cos\theta)n n^t+\sin \theta \begin{pmatrix} 0 & -n_3 & n_2\\ n_3 & 0 & -n_1\\ -n_2 & n_1 & 0 \end{pmatrix}$$ being $n$ the vector of the axis of rotation and $\theta$ the angle of rotation, which in this case would be $n$ the the vector of the subspace and $\theta=180^\circ$?
One simple way to solve the first part is to find a basis $f_1,f_2$ of $V_1$ and consider $f_3 = f_1 \times f_2.$
Then define a linear map $T$ by $$ T(f_1) = f_1, T(f_2) = f_2 \text{ and } T(f_3) = - f_3.$$ Now you only need to find the matrix of $T$ with respect to the standard basis.