Systems of Linear Equations

A linear system is compactly $A\mathbf{x} = \mathbf{b}$. Row picture: each equation is a hyperplane; solutions are intersections. Column picture: we ask whether $\mathbf{b}$ is a linear combination of the columns of $A$, i.e., whether $\mathbf{b}$ lies in the column space. The column picture is the one that generalizes (least squares, regression, factor models all live there).