I'm currently hopelessly stuck on an exercise in linear algebra and I could use some hints.
Let $V$ be a finite-dimensional vector space over the reals and let $G$ be a finite subgroup of $$\operatorname{GL}(V):=\{f:V\to V\,|\,f\text{ is linear and bijective}\}.$$ Prove: There exists an inner product $\langle\cdot,\cdot\rangle:V\times V\to\mathbb R$ under which all $f\in G$ are isometries, i.e. $$\langle f(\mathbf v),f(\mathbf w)\rangle=\langle\mathbf v,\mathbf w\rangle\quad\forall\ f\in G,\mathbf v,\mathbf w\in V.$$