Lets say we have the following Matrix
$$\left[
\begin{matrix}
2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 0 \\
\end{matrix}
\right]$$

Obviously a basis for this would be 
$$\begin{Bmatrix}
\left[
\begin{matrix}
2\\
0\\
0\\
\end{matrix}
\right]
,\left[
\begin{matrix}
0\\
2\\
0\\
\end{matrix}
\right]
\end{Bmatrix}$$
But would another valid basis be
$$\begin{Bmatrix}
\left[
\begin{matrix}
1\\
0\\
0\\
\end{matrix}
\right]
,\left[
\begin{matrix}
0\\
1\\
0\\
\end{matrix}
\right]
\end{Bmatrix}$$