How to remove space from mathematical formulas?

One way to reduce the spacing between the columns is to use @{}

I've put @{} between every column, but of course you can use it just on the columns that you wish.

You can think of the @{<stuff>} operation as adding <stuff> to every element in that column.

\documentclass{article}


\begin{document}

Original
\begin{equation}
  \begin{array}{llllll}
    f_1( & v_1, & v_2  &     &     & ) = 0 \\
    f_2( &      & v_2, & v_3 &     & ) = 0 \\
    f_3( &      & v_2, &     & v_4 & ) = 0 \\
    f_4( & v_1, &      &     & v_4 & ) = 0 
  \end{array}
\end{equation}

New
\begin{equation}
  \begin{array}{@{}l@{}l@{}l@{}l@{}l@{}l@{}}
    f_1( & v_1, & v_2  &     &     & ) = 0 \\
    f_2( &      & v_2, & v_3 &     & ) = 0 \\
    f_3( &      & v_2, &     & v_4 & ) = 0 \\
    f_4( & v_1, &      &     & v_4 & ) = 0 
  \end{array}
\end{equation}

\end{document}

Suggested by @egreg: you can use this idea to put \, after columns 2, 3, and 4, which gives

\begin{equation}
  \begin{array}{@{}l@{}l@{\,}l@{\,}l@{\,}l@{}l@{}}
    f_1( & v_1, & v_2  &     &     & ) = 0 \\
    f_2( &      & v_2, & v_3 &     & ) = 0 \\
    f_3( &      & v_2, &     & v_4 & ) = 0 \\
    f_4( & v_1, &      &     & v_4 & ) = 0 
  \end{array}
\end{equation}

As @barbarabeeton pointed out, it doesn't seem necessary to use 6 columns; 5 columns will work ok

\begin{equation}
  \begin{array}{l@{\,}l@{\,}l@{\,}l@{}l@{}}
    f_1( v_1, & v_2  &     &     & ) = 0 \\
    f_2(      & v_2, & v_3 &     & ) = 0 \\
    f_3(      & v_2, &     & v_4 & ) = 0 \\
    f_4( v_1, &      &     & v_4 & ) = 0 
  \end{array}
\end{equation}

I assume you have the last column empty so that the closing ) will be aligned.