How to reproduce a matrix with vectors labels?

Question

I would like to write a matrix like the picture below (that is my target): But I'm struggling with f1 f2...f(e1) f(e2).

$\begin{pmatrix}
a_{1,1}      &      & a_{1,j}   & \dots & a_{1,p} \\
a_{2,1}      &      & a_{2,j}   & \dots & a_{2,p} \\
\vdots        &         & \vdots    & \ddots& \vdots \\
a_{n,1}      & \dots    & a_{n,j}   & \dots & a_{n,p}\\
\end{pmatrix}$

Answer 1

You can use \bordermatrix.

\documentclass{article}

\begin{document}
\[
  \bordermatrix{
           & f(e_1)  & \dots & f(e_j)  & \dots  & f(e_p)  \cr
    f_1    & a_{1,1} &       & a_{1,j} & \dots  & a_{1,p} \cr
    f_2    & a_{2,1} &       & a_{2,j} & \dots  & a_{2,p} \cr
    \vdots & \vdots  &       & \vdots  & \ddots & \vdots  \cr
    f_n    & a_{n,1} & \dots & a_{n,j} & \dots  & a_{n,p} \cr
  }
\]
\end{document}

Answer 2

A solution based on gauss:

\documentclass{article}
\usepackage[x11names]{xcolor}
\usepackage{mathtools}
\usepackage{gauss, array, pgffor}

\def\matlab{\mult}
\newenvironment{annotmatrix}[1][]{%
\let\rowmultlabel\text \let\colmultlabel\text\setlength{\rowarrowsep}{0pt}\setlength{\colarrowsep}{0.8ex}%
\gmatrix[#1]}
{\endgmatrix}

\begin{document}

\[\renewcommand\arraystretch{1.3}
\begin{tabular}{ >{\footnotesize\color{PaleGreen3}$}c<{\!$}@{}}
f_1 \\[-0.8ex] f_2 \\ \vdots\\[-0.1ex] f_n
\end{tabular}
  \begin{annotmatrix}[p]%
    a_{11} & \dots & a_{1j} & \dots &a_{1p} \\%
    a_{21} & \dots & a_{2j} & \dots &a_{2p} \\%
     \vdots & & \vdots & &\vdots \\%
    a_{n1} & \dots & a_{nj} & \dots &a_{np}
    \colops
\foreach \i/\label in {0/ f({\color{RoyalBlue3}e_1}), 1/\dotsm, 2/ f({\color{RoyalBlue3}e_j}), 3/\dotsm, 4/ f({\color{RoyalBlue3}e_p})}{\matlab{\i}{\footnotesize$ \label$}}
  \end{annotmatrix}
\]

\end{document}