Mathematical notation for $i^{th}$ canonical basis vector for the n-dimensional space

Question

One of my constraints in an optimization problem involves using canonical basis vector for the n-dimensional space.

How do I precisely write $j^{th}$ canonical basis vector for the n-dimensional space as a vector.

Normally its like

$n\times 1$ vector $\mathbf{e}_j$ is

\begin{gather*} \mathbf{e}_j=\begin{cases} 1 & \text{, } i=j\\ 0 & \text{, } i\neq j\\ \end{cases} \end{gather*}

and $1\leq j\leq n$.

Answer 1

I think what you’ve already written basically suffices, but to be a bit clearer you could say:

Assuming you’re working in the real numbers...

The $j^{th}$ canonical basis vector $\mathbf{e}_j$ = $\begin{pmatrix}a_1\\a_2\\\vdots\\a_n\end{pmatrix} \in \mathbb{R}^n$ where $a_i = 1$ if $i = j$ and $0$ otherwise.