Column Space

When choosing a specific $A$, the system $Ax=b$ has only solutions $x=\left[\begin{array}{c}x\_1x\_2⋮x\_n\end{array}\right]$ when $b$ lies inside the column-space of $A$, i.e. the subspace of $R^n$ that the independent vectors of $A$ collectively define. This is the column space of $A$, i.e. the subspace that is spanned by all linear combinations of the rows of $A$ for a particular condition $Ax=b$.