In the book Linear Algebra Done Right (Axler, 237), the following is the statement of the theorem that gives us the existence of a singular value decomposition:
7.51 $\qquad$ Singular Value Decomposition
Suppose $T\in\mathcal{L}(V)$ has singular values $s_{1},\ldots,s_{n}$. Then there exist orthonormal bases $e_{1},\ldots,e_{n}$ and $f_{1},\ldots,f_{n}$ of $V$ such that
$$Tv=s_{1}\langle v,e_{1} \rangle f_{1}+\cdots+s_{n}\langle v,e_{n} \rangle f_{n}$$
for every $v\in V$.
Now, I have seen elsewhere that we can do this for general rectangular matrices, yet this statement of the theorem seems to indicate that it can only be done for square matrices (since it says $T\in \mathcal{L}(V)$ rather than $T\in \mathcal{L}(V,W)$.
Is it enough to know that you can do it to a square matrix to know that you can do it to a rectangular matrix, since you can add enough rows/columns of all $0$'s to make a rectangular matrix square? If so, why does doing this not affect the decomposition? I feel like I'm missing something obvious, but I appreciate any insight on the matter.