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Exercises 3.2 Exercises
Exercise Group. In each of the following determine whether a given subset \(S\) is linearly dependent or independent.
1. \(V=\R^2\) as a vector space over \(\R\) and \(S=\{(1,1),(1,-1)\}\text{.}\) 2. \(V=\R^3\) as a vector space over \(\R\text{,}\) and \(S=\{(1,1,1),(2,1,1),(-1,1,-1),(7,7,7)\}\text{.}\) 3. \(2\times2\) 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*}
4. \(V\) to be the vector space over a field \(F\) of all polynomials in variable \(x\) of degree at most \(n\text{,}\) and \(S=\{1,x,x^2,\ldots,x^n\}\text{.}\) 5. \(V=\End_{\R}(\R)\) over \(\R\) (refer to Example 2.2.3 ), and \(S=\{\ell_{\pi},\ell_{3}\}\text{.}\)
