Identity Matrix and Standard Basis Vectors

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, with real matrices and their entries, and points of $\mathbb{R}^{n}$ with components as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii).

(i) The identity matrix $I_{n}\in\mathcal{M}_{n}(\mathbb{R})$ is the matrix $E$ of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (v); its entries are written $\delta_{ij}=(I_{n})_{ij}$, so that $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ if $i\neq j$ ($i,j\in[n]$). It is not the constant map $1$ of Pointwise Operations on Real-Valued Maps (ii).

(ii) For $k\in[n]$, the $k$-th standard basis vector $e_{k}\in\mathbb{R}^{n}$ is the point $e$ of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (vi), so that $(e_{k})_{i}=\delta_{ik}$ for $i\in[n]$.