Just a quick check, I am asked to determine for which values of the parameter $k$ does this set of vectors $\{(-1, k, -1),\ (2, 2, 0),\ (1, k, 1)\}$ span $\mathbb{R}^3$.
My reasoning is that for them to span $\mathbb{R}^3$ they have to be linearly independent, therefore if we represent them as column vectors multiplied by some arbitrary scalars $c1$ - $c3$ respectively and add them together to equal the $0$ vector in $\mathbb{R}^3$ we should be able to determine if they are linearly independent or not by seeing if there aren't any non-trivial solutions to this system of equations.
Hope I'm right, thanks for the help.
You are right. And your reasoning is equivalent to find the value(s) of $k$ for which the determinant of the matrix that has as columns the three vectors is not null. Do you see why?