This is from Trefethen and Bau, page 33, theorem 5.2. SVD of $A=U.\sum.V^*$ where $U,V$ are unitary and $\sum$ is diagonal. It states that range$(A) =\langle u_1,...,u_r \rangle$ and null $(A) = \langle v_{r+1},...,v_n \rangle$.
$r$ is the rank of $A$.
The proof then states that this theorem is the consequence of range $(\sum) = \langle e_1,...,e_r \rangle$ and null $(\sum) = \langle e_{r+1},...,e_n \rangle$.
I get the range and null of $\sum$, intuitively. I don’t understand how the theorem follows from that statement.