Determine whether the set c=\left\{\left [ \begin{matrix} 1 \\ 1 \end{matrix} \right ] ,\left [ \begin{matrix} 1 \\ 0 \end{matrix} \right ] ,\left [ \begin{matrix} 2 \\ 2 \end{matrix} \right ] \right\} is linearly independent or linearly dependent.
Question 1.2.7: Determine whether the set c= {[1 1], [1 0], [2 2]} is linear...
We consider the equation
\left [ \begin{matrix} 0 \\ 0 \end{matrix} \right ] =c_{1}\left [ \begin{matrix} 1 \\ 1 \end{matrix} \right ] +c_{2}\left [ \begin{matrix} 1 \\ 0 \end{matrix} \right ] +c_{3}\left [ \begin{matrix} 2 \\ 2 \end{matrix} \right ]We observe that taking c_{1}=-2, \ \ c_{2}=0, \ \ and c_{3}=1 satisfies the equation. Hence, by definition, the set C is linearly dependent.
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