Euclidean vector spaces/Matrix space and trace/Exercise

Let ${}V=\operatorname {Mat} _{n}(\mathbb {R} )$. Show that ${}V$, endowed with the mapping

$\left\langle -,-\right\rangle \colon V\times V\longrightarrow \mathbb {R} ,(A,B)\longmapsto \operatorname {trace} {\left(B^{t}A\right)},$

is a Euclidean vector space.

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