The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $m,n,p\in\mathbb{N}$, with real matrices and their entries, points of $\mathbb{R}^{n}$ and their components as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §membership, and sums as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings.

¶ There is exactly one map $\pi:\mathcal{M}_{m\times n}(\mathbb{R})\times\mathcal{M}_{n\times p}(\mathbb{R})\to\mathcal{M}_{m\times p}(\mathbb{R})$ such that $\pi(A,B)_{ik}=\sum_{j=1}^{n}A_{ij}B_{jk}$ for all $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $B\in\mathcal{M}_{n\times p}(\mathbb{R})$, $i\in[m]$ and $k\in[p]$.

¶ There is exactly one map $\phi:\mathcal{M}_{m\times n}(\mathbb{R})\times\mathbb{R}^{n}\to\mathbb{R}^{m}$ such that $\phi(A,x)_{i}=\sum_{j=1}^{n}A_{ij}x_{j}$ for all $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $x\in\mathbb{R}^{n}$ and $i\in[m]$.

¶ There is exactly one map $\tau:\mathcal{M}_{m\times n}(\mathbb{R})\to\mathcal{M}_{n\times m}(\mathbb{R})$ such that $\tau(A)_{ji}=A_{ij}$ for all $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $i\in[m]$ and $j\in[n]$.

¶ There is exactly one map $t:\mathcal{M}_{n}(\mathbb{R})\to\mathbb{R}$ such that $t(A)=\sum_{i=1}^{n}A_{ii}$ for all $A\in\mathcal{M}_{n}(\mathbb{R})$.

¶ There is exactly one $E\in\mathcal{M}_{n}(\mathbb{R})$ such that, for all $i,j\in[n]$, $E_{ij}=1$ if $i=j$ and $E_{ij}=0$ if $i\neq j$.

¶ For every $k\in[n]$ there is exactly one $e\in\mathbb{R}^{n}$ such that, for all $i\in[n]$, $e_{i}=1$ if $i=k$ and $e_{i}=0$ if $i\neq k$.