Proof of Theorem MVSLD

Form the $m\times n$ matrix $A$ whose columns are $\vect{u}_i$, $1\leq i\leq n$. Consider the homogeneous system $\homosystem{A}$. By Theorem HMVEI this system has infinitely many solutions. Since the system does not have a unique solution, Theorem LIVHS says the columns of $A$ form a linearly dependent set, as desired.