In the SVD of $A = U \Sigma V^T$, how does one know that V and U actually span the column and row space of A (respectively for each one)?
I do know how to find such a U and V and $\Sigma$ by just using the matrices $AA^T$ and/or $A^TA$ and then solving for the eigenvalues/vectors and then plugging back to $AV = U \Sigma$ to find a the appropriate vectors that satisfy the factorization.
However, it is not clear to me at all how one would show/proof that U actually spans the column space of A $C(A)$ and that V spans the row space $C(A^T)$. How does one show that rigorously? Or is there an intuitive way of seeing how this is obvious? An answer with both would be great!
Is it suppose to be obvious? Because I've noticed that people talk about it as something obvious. Maybe its just very well known that its taken for granted? Or maybe there a very intuitive way of understanding this and thus, it should be obvious?