\begin{tikzpicture}[scale=1.2]

  %  \draw[step=0.1, color=lightgray] (-5,-2.5) grid(7,5);

    \draw[-triangle 45] (-4.5,0) -- (6.5,0) node[right] {$x$};

     \foreach \x in {-4,-2,2,4,6}
       \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};

     \draw[-triangle 45] (0,-2.5) -- (0,4) node[above] {$y$};
         \foreach \y in {-2,2}
         \draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};



    \draw [color=red, thick, -triangle 45](0,0) -- (2,1) node[midway,sloped,above] {$\vec a$};
    \draw [color=red, dashed, thin] (2,-0.1) -- (2,1);
    \draw [color=red, dashed, thin] (-0.1,1) -- (2,1);
        \node[red] at (2,-0.5) {$a_1$};
        \node[red, left] at (-0.1, 1) {$a_2$};


    \draw [color=green!65!black, thick, -triangle 45](0,0) -- (-4,-2) node[midway,sloped,above] {$-2 \cdot \vec a$};
    \draw [color=green!65!black, dashed, thin] (-4,0.1) -- (-4,-2);
    \draw [color=green!65!black, dashed, thin] (0.1,-2) -- (-4,-2);
            \node[green!65!black, right] at (0.25,-2) {$-2\cdot a_2$};



    \draw [color=blue, thin, -triangle 45, dashed](0,0) -- (6,3) node[midway,sloped,above] {$3 \cdot \vec a$};
    \draw [color=blue, dashed, thin] (6,-0.1) -- (6,3);
    \draw [color=blue, dashed, thin] (-0.1,3) -- (6,3);
     \node[blue] at (6,-0.5) {$3 \cdot \vec a_1$};
     \node[blue,left] at (-0.1,3) {$3 \cdot \vec a_2$};

\draw (-0.5,2.5) node[left, text width=4cm, color=green!65!black]{$k<0$: Richtung \"andern und Strecken/Stauchen um Faktor $-k$ \underline{hier: $-k=2$}};


\draw (1.5,3.2) node[right, text width=4cm, color=blue]{$k>0$: Strecken/Stauchen um Faktor k \underline{hier: $k=3$}};


\end{tikzpicture}