Originally Posted by
Ackbeet Right-ho. Then the way I would go about it is this: let the columns of $\displaystyle A$ be denoted $\displaystyle \mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{5} .$ They are column vectors in $\displaystyle \mathbb{R}^{4}.$ Let
$\displaystyle \mathbf{r}=\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{ 4}\end{bmatrix}$
be an arbitrary vector in $\displaystyle \mathbb{R}^{4}.$ We want to know if there is a set of scalars, $\displaystyle b_{1},b_{2},\dots,b_{5},$ such that
$\displaystyle b_{1}\mathbf{a}_{1}+b_{2}\mathbf{a}_{2}+\dots+b_{5 }\mathbf{a}_{5}=\mathbf{r},$
right? Now, this equation that I've just written down can be translated into a system of equations for the unknown constants $\displaystyle b_{j}.$ Your goal is to determine the rank of the coefficient matrix for that system. If the rank is 4, then you've got yourself a spanning set with those $\displaystyle \mathbf{a}_{j}$'s. Otherwise, if the rank is less than 4, you don't. Make sense? Where would you go next?