Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$? Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$?

You have this coefficient(augmented) matrix: $\begin{pmatrix}
2&-1&1&3\\
-1&1&1&-1\\
3&1&9&0\\
2&-3&-5&-1
\end{pmatrix}$
Reduce it to the row echelon form and check whether it is consistent.
You basically need to check whether after row reduction its last non-zero row has a pivot in the last column or not.If it has in the last column then the equation set is inconsistent so you dont have a solution.Else you have solution(infinitely many) implying that the vector belongs to the span of the given three vectors.
