Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. The trace of any matrix $M$ of $\mathcal{M}_{n}(\mathbb{R})$ is denoted $\operatorname{tr}(M)$.
Show that the map defined on $\mathcal{M}_{n}(\mathbb{R}) \times \mathcal{M}_{n}(\mathbb{R})$ by $(A, B) \mapsto \langle A \mid B \rangle = \operatorname{tr}(A^{\top} B)$ is an inner product on $\mathcal{M}_{n}(\mathbb{R})$.