Let $S = \{(0,0), (0,1), (1,0), (1,1)\} \subset \mathbb{R}^2$ and consider the vector space $\mathbb{R}^S$.
Show that if
$f_1(m, n) = \begin{cases} 1 & (m, n) = (0, 0) \\ 0 & (m, n) \neq (0, 0) \end{cases}$
$f_3(m, n) = \begin{cases} 1 & (m, n) = (1, 0) \\ 0 & (m, n) \neq (1, 0) \end{cases}$
the set $\{f_1, f_2, f_3, f_4\}$ is a basis for $\mathbb{R}^S$.
$f_2(m, n) = \begin{cases} 1 & (m, n) = (0, 1) \\ 0 & (m, n) \neq (0, 1) \end{cases}$
$f_4(m, n) = \begin{cases} 1 & (m, n) = (1, 1) \\ 0 & (m, n) \neq (1, 1) \end{cases}$
