\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}

\usepackage{amsmath}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc,matrix}

\begin{document}
\begin{preview}
\begin{tikzpicture}
    \begin{axis}[
        width=10cm, height=10cm,     % size of the image
        axis lines=middle,
        grid = major,
        grid style={dashed, gray!30},
        %xmode=log,log basis x=10,
        %ymode=log,log basis y=10,
        xmin=-100,     % start the diagram at this x-coordinate
        xmax= 100,     % end   the diagram at this x-coordinate
        ymin=-100,     % start the diagram at this y-coordinate
        ymax= 100,     % end   the diagram at this y-coordinate
        /pgfplots/xtick={-100,-80,...,100},
        /pgfplots/ytick={-100,-80,...,100},
        tick align=outside,
        legend pos= north west,
        legend cell align=left,
        legend style={legend pos=north west,font=\tiny}]

      \addplot[domain=-100:100, red, thick]  {x};
      \addlegendentry{$\begin{pmatrix}0\\0\end{pmatrix} + U$};

      \addplot[domain=-100:100, blue, thick] {x + 12};
      \addlegendentry{$\begin{pmatrix}0\\12\end{pmatrix} + U$};

      \addplot[domain=-100:100, purple, thick] {x - 42};
      \addlegendentry{$\begin{pmatrix}0\\-42\end{pmatrix} + U$};

      \addplot[domain=-100:100, lime, thick] {x + 50};
      \addlegendentry{$\begin{pmatrix}0\\50\end{pmatrix} + U$};
    \end{axis}
\end{tikzpicture}
\end{preview}
\end{document}