Let \(B=(E_{11}, E_{12}, E_{21}, E_{22})\) be the standard basis of \(M_{22}\text{.}\) Apply \([\phantom{\boldv}]_B\) to the elements of the given \(S\) to get a corresponding set \(S'\subseteq\R^4\text{:}\)
\begin{equation*}
S'=\left\{ [A_1]_B=(2,1,0,-2), [A_2]_B=(1,1,1,-1), [A_3]_B=(0,1,2,0), [A_4]_B=(-1,0,1,1) \right\}\text{.}
\end{equation*}
Apply the column space procedure of
ProcedureΒ 3.8.13 to contract
\(S'\) to a basis
\(T'\) of
\(\Span S'\text{.}\) This produces the subset
\begin{equation*}
T'=\{[A_1]_B=(2,1,0,-2), [A_2]_B=(1,1,1,-1)\}
\end{equation*}
Translating back to \(V=M_{22}\text{,}\) we conclude that the corresponding set
\begin{equation*}
B=\{A_1, A_2\}
\end{equation*}
is a basis for \(W'=\Span S\text{.}\) We conclude that \(\dim W'=2 \text{.}\)