Let $W$ and $T$ be linear spaces over $\mathbb F$, with $n=\dim(W)$ and $t=\dim(T)$. Let $U_1,\ldots,U_m$ be a family of subspaces of $W$ with $\dim(U_i)=r_i$ for each $i$.
The linear map $\phi:W\to T$ is defined to be in general position (with respect to the $U_i$) iff $\dim(\phi(U_i))=\min\{r_i,t\}$ for each $i$. In other words, $\dim(\phi(U_i))$ is the maximum possible for each $i$.
The theorem states that given a large enough field, such a linear map always exists. Formally, if the field $\mathbb F$ has order $> (n-t)(m+1)$, then a linear map $\phi:W\to T$ in general position always exists.
Does anyone know of a reference to the proof of this theorem? I found it in the textbook "Linear Algebra Methods in Combinatorics" by Babai and Frankl, Theorem 3.13, page 63. (The textbook is easily found online.) However, the proof seems to be wrong, specifically Proposition 3.9, whose statement seems to be false... or maybe I'm confused?
Edit: Resolved, thanks! See the comments.