Chapter 5.5, Problem 12E

In Exercises $8-13$, a subset $S$ of vector space $V$ is given. In each case choose one of the statements $i),ii),$ or $iii)$ that holds for $S$ and verify that this is the case.

$i)$ $S$ is a basis for $V$.

$ii)$ $S$ does not span $V$.

$iii)$ $S$ is linearly dependent.

$12$. $S=\left\{\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]\right\};$

$V$ is the set of all $\left(2\times 2\right)$ real matrices.