$$Av_i= \begin{cases} \sigma_iu_i & i = 1, \ldots , r \\ 0 & i = r+1, \ldots , m \end{cases}$$
$$A^Tu_i= \begin{cases} \sigma_iv_i & i = 1, \ldots , r \\ 0 & i = r+1, \ldots , m \end{cases}$$
1.) Show that the top equation implies the matrix equation $AV = U\Sigma$.
2.) Show that the bottom equation implies the matrix equation $A^TU = V\Sigma^T$
3.) Show that either one implies the SVD $A=U\Sigma V^T$
I understand the implications of both equations, however, I'm not sure how to show it in a simplistic way.