Multiple Basis for a Single Vector Space

A single vector space can have multiple basis.

Example

$\{(1,0),(0,1)\}$ and $\{(1,1),(1,-1)\}$ are bases of ${\mathbb{R}}^2$ Proof: Linear independence: Suppose $x(1,1)+y(1,-1)=(0,0)$

$$\Rightarrow \left[\begin{array}{ccc}1& 1& 0\\ 1& -1& 0\end{array}\right]\stackrel{(1)(R1+R2\to R2)}{\to }\left[\begin{array}{ccc}1& 1& 0\\ 2& 0& 0\end{array}\right]\stackrel{\frac{1}{2}(R2)\to R2}{\to }\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& 0\end{array}\right]\stackrel{-1(R2)\to R1}{\to }$$

Note that $(1,1)$ and $(1,-1)$ also span ${\mathbb{R}}^2$