Determine whether the list $v_1 = (1, 1, −1, 2), v_2 = (2, 2, −3, 1), v_3 = (−1, −1, 0, −5)$ in $k^4$ is linearly independent.
Work: I put it into $Ax=0$ form: \begin{bmatrix} 1 & 2 & -1 & | & 0 \\ 1 & 2 & -1 & | & 0 \\ -1 & -3 & 0 & | & 0 \\ 2 & 1 & -5 & | & 0 \end{bmatrix} and reduced it down to: \begin{bmatrix} 1 & 0 & -3 & | & 0 \\ 0 & 0 & 0 & | & 0 \\ 0 & 1 & 1 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} and so $Rank (A)=2 \neq 4$.
I'm not sure what to do from here - does this mean the list is linearly dependent?
They are linearly dependent as you see, we have here three equations in four variables so we can always find solutions where all four are not zero. Hence the system is linearly dependent.