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*} Let $W=\spn{S}$ be the span of S. Is the vector $\colvector{-1\\8\\-4}$ in $W$? Give an explanation of the reason for your answer.
2. Use $S$ and $W$ from the previous question. Is the vector $\colvector{6\\5\\-1}$ in $W$? Give an explanation of the reason for your answer.
3. For the matrix $A$ below, find a set $S$ so that $\spn{S}=\nsp{A}$, where $\nsp{A}$ is the null space of $A$. (See Theorem SSNS.) \begin{equation*} A= \begin{bmatrix} 1 & 3 & 1 & 9\\ 2 & 1 & -3 & 8\\ 1 & 1 & -1 & 5 \end{bmatrix} \end{equation*}