Note that the vector space V is finite-dimensional.
So basically, we need to rewrite $T\in\mathcal{L}(V)$ as $T=a(S_1+S_2)$.I thought of using polar decomposition to write $T=G\sqrt{T^*T}$, but I'm not sure how to transform that into a sum of isometries multiplied by a constant. The same goes for singular value decomposition, $$Tv=s_1 \langle v,e_1 \rangle f_1+...+s_n\langle v,e_n \rangle f_n$$ where $e_1,...,e_n$ and $f_1,...,f_n$ are orthonormal bases of $V$, $s_1,...,s_n$ are singular values of $T$, and $v\in V$. Not sure where to go from there.