I am currently struggling to understand the proof for SVD.
I don't understand why:
$u_1, \ldots ,u_r$ is an orthonormal basis for the column space,
$u_{r+1}, \ldots ,u_m$ is an orthonormal basis for the left nullspace $N(AT),$
$v_1, \ldots , v_r$ is an orthonormal basis for the row space,
$v_{r+1}, \ldots , v_n$ is an orthonormal basis for the nullspace $N(A).$
I'm following the proof on Strang's proof of SVD and intuition behind matrices $U$ and $V$, using the reply by Matthew Leingang.
Also curious why we are initially only dealing with the $v_1,\ldots,v_r$ before joing the remaining vectors after.
I'd appreciate if someone could post a full detailed proof on it as I'm struggling to understand other resources.
Thanks