Exercises 3.2 Exercises
In each of the following determine whether a given subset \(S\) is linearly dependent or independent.
1.
Consider \(V=\R^2\) as a vector space over \(\R\) and \(S=\{(1,1),(1,-1)\}\text{.}\)2.
Consider \(V=\R^3\) as a vector space over \(\R\text{,}\) and \(S=\{(1,1,1),(2,1,1),(-1,1,-1),(7,7,7)\}\text{.}\)3.
Consider the set of \(2\times2\) matrices over \(\C\text{,}\) \(M_{2}(\C)\) as a vector space over \(\C\text{,}\) and
\begin{equation*}
S=\bigg\{\begin{pmatrix}1\amp 0\\0\amp 0\end{pmatrix},\begin{pmatrix}0\amp 1\\0\amp 0\end{pmatrix},\begin{pmatrix}0\amp 0\\1\amp 0\end{pmatrix},\begin{pmatrix}0\amp 0\\0\amp 1\end{pmatrix}\bigg\}\text{.}
\end{equation*}