Add description next to matrix

Question

I have the following definition of a standard basis vector:

This text was compiled by the following code:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

For $ \mathbb{R}^n $, we let $e_j \in \mathbb{R}^n$ (the standard basis vector) 
denote the vector with a 1 in the $j$-th entry and 0 elsewhere: 
\[
    e_j = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 (\text{jth spot}) \\ 0 \\ \vdots \\ 0 \end{pmatrix}
.\] 

\end{document}

But I don't like the way the entry marked as the "jth spot" comes out. I would like some way to maybe take that description out and put it next to it outside of the matrix. How could I do this?

Edit: I know that, in this case, it is possible to do this just by writing j\text{th spot} right next to the matrix, since the 1 happens to be in the center of the matrix. However, I don't like this solution because it doesn't generalize to other spots in the matrix. What other ways are there to do this?

Answer 1

You can set the element in a separate array to the right of the pmatrix. Just add the same number of lines and a closing element to avoid gobbling of the closing \\ (you'll see \mathstrut in my example). For completeness, I've added an "i-th element" label as well, so you can see what that looks like

\documentclass{article}

\usepackage{amsmath,amssymb}

\begin{document}

For $\mathbb{R}^n$, we let $e_j \in \mathbb{R}^n$ (the standard basis vector) denote the vector with a~1 in the $j$-th entry and~0 elsewhere: 
\[
  e_j = \begin{pmatrix}
    0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0
  \end{pmatrix}~
  \begin{array}{@{} c @{}}
     \\ \\ (i\text{-th entry}) \\ (j\text{-th entry}) \\ \\ \\ \mathstrut
  \end{array}
\] 

\end{document}

Answer 2

Like this?

With nicematrix package is simple:

\documentclass{article}
\usepackage{nicematrix}

\begin{document}

\[
    e_j = \begin{pNiceMatrix}[last-col,nullify-dots]
    0       \\ 
    \vdots  \\ 
    0       \\  
    1       &   (j\text{th spot}) \\ 
    0       \\ 
    \vdots  \\ 
    0 
\end{pNiceMatrix}
\]

\end{document}

For showed result you need to compile this MWE at least three times since nicematrix is based on tikz package with options remember picture and overlay.

Edit:
Considered are @egreg comments.