Let \(V\) be an \(n\)-dimensional vector space and \(B\) a basis. Prove that the map \(F:\text{End}[(V) \to M_{n}(CC)\) given by \(F(T) = [T]_{B}\) is an isomorphism of unital rings.

Prove that \((AB)^* = B^*A^*\).

Prove that \(\text{Tr}(AB) = \text{Tr}(BA)\).

Let \(V\) be an inner product space. Show that \(U \in \text{End}(V)\) is unitary if and only if \(\left\| Uv \right\| = \left\| v \right\|\) for all vectors \(v \in V\). (Hint: use polarization formula below).

Prove the polarization formula for complex inner product spaces: \[ \langle v,w \rangle = \frac{1}{4} \left[ \left\| v+w \right\|^2 - \left\| v - w \right\|^2 + \i\left\| v + iw \right\|^2 - \i\left\| v-iw \right\|^2\right] \]