Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality.

Does there exist a bijection $f : B_V \to B_W$ such that, for each $b_V \in B_V$, the coefficient of $f(b_V)$ in $T(b_V)$ is nonzero?

If $V,W$ are finite-dimensional, the answer is yes: $\det(T) \ne 0$, hence some monomial term is nonzero. This exhibits a satisfactory $f$. Alternately, build $f$ inductively by Laplace expansion along a row or column. (Some term $a_{ij} \cdot \text{(complementary minor)}$ is nonzero, and so on.)

Does this hold in general? I have tried using Zorn's lemma, but it seems tricky enough that maybe I'm overlooking a straightforward counterexample.

I can state what I've attempted if there's interest.

(I have tagged this as linear and homological algebra since my desired application is in the latter.)