<aside>

Given an <m>m\times n</m> real matrix <m>A</m>, show that
there exist (appropriately sized) elementary matrices
<m>U</m>,<m>V</m> so that <m>UAV</m> has the form 
<me>
    UAV =
        \begin{bmatrix} 
            I_r\amp \mathbf 0\\
            \mathbf 0\amp\mathbf 0
        \end{bmatrix}.
</me>
where <m>I_r</m> is the <m>r\times r</m> identity matrix, 
<m>r</m> the rank of <m>A</m>, and the other entries are
all zeros.
 
<aside>
<p>
    Note that when we work with modules over a PID instead of
    vector spaces over a field, this construct leads to a
    diagonal matrix called the
    <term>Smith normal form</term> of the matrix <m>A.</m>
</p>
<p>
    The nonzero diagonal elements of the Smith normal form are
    called the <term>invariant factors</term>.
</p>
</aside>