A nxn, x and B are both nx1 Ax=B
First show that if rows are all independent then column are all independent:
Intuitively: n independent rows (representing n “non-parallel” equations) in n unknowns means the system of equations has a unique solution (the planes or whatever intersect in one point) The same thing stated from the point of view of the column space: there is only one possible coordinate for B using the columns of A as a coordinate system. If there is only one possible coordinate for B using column vectors of A then those column vectors must be linearly independent.(obvious no?) From here its easy to generalize to cases where some rows are dependent as well, just take the smallest square matrices that has only independent rows and then add the dependent row...? Not rigorous (almost though) but this really explains it no?