1. Let $S$ be the set of three vectors below. \begin{equation*} S=\set{\colvector{1\\2\\-1},\,\colvector{3\\-4\\2},\,\colvector{4\\-2\\1}} \end{equation*} Is $S$ linearly independent or linearly dependent? Explain why.
2. Let $S$ be the set of three vectors below. \begin{equation*} S=\set{\colvector{1\\-1\\0},\,\colvector{3\\2\\2},\,\colvector{4\\3\\-4}} \end{equation*} Is $S$ linearly independent or linearly dependent? Explain why.
3. Is the matrix below singular or nonsingular? Explain your answer using only the final conclusion you reached in the previous question, along with one new theorem. \begin{equation*} \begin{bmatrix} 1 &3 &4\\ -1 &2 &3\\ 0 &2 &-4 \end{bmatrix} \end{equation*}