\begin{itemize}
\item[(a)] Show that for all $u, v \in \mathbb{R}^{d}$ and $A \in \mathscr{M}_{d}(\mathbb{R})$, we have $\langle u, Av \rangle_{\mathbb{R}^{d}} = \langle uv^{T}, A \rangle$.
\item[(b)] Show that $\operatorname{tr}(AB) = \operatorname{tr}(BA)$ for $A, B \in \mathscr{M}_{d}(\mathbb{R})$.
\item[(c)] Deduce that for all $A, B$ and $C$ in $\mathscr{M}_{d}(\mathbb{R})$ we have
$$\langle A, BC \rangle = \langle B^{T} A, C \rangle = \langle AC^{T}, B \rangle.$$
\end{itemize}