I am really stuck on exercise 5 in chapter 1.3 of Greub's book Multilinear Algebra.
The exercise asks me to show that if $\varphi \colon E \times F \to G$ and $\psi \colon E \times F \to H$ are bilinear maps such that $N_1(\varphi) \subseteq N_1(\psi)$ and $N_2(\varphi) \subseteq N_2(\psi)$, then there exists a linear map $f \colon G \to H$ such that $\psi = f \varphi$.
The definition of $N_1$ and $N_2$ are: $N_1(\varphi) = \{ x \in E \mid \varphi(x, y) = 0 \; \forall y \in F \}$, and $N_2(\varphi) = \{ y \in F \mid \varphi(x, y) = 0 \; \forall x \in E \}$.
It is clear to me that $N_1(\varphi)$ is the kernel of the "adjoint" linear map $E \to \mathrm{Hom}(F,G)$, and $N_2(\varphi)$ the kernel of the map $F \to \mathrm{Hom}(E, G)$. But I don't see how this is helpful.
Another problem is that assuming the exercise to be true I should at the very least be able to show that if $\varphi(x_1, y_1) = \varphi(x_2, y_2)$, for some $(x_1, y_1), (x_2, y_2) \in E \times F$, then $\psi(x_1, y_1) = \psi(x_2, y_2)$ as well. But even this I have no idea how to show.
Any hints or help would be greatly appreciated.
Different edition from mine...
ERRATA list. Each listing has (p,n) where p is the page number and : postive n means n lines from the top down; n negative means that many lines up from the bottom of the page.
https://github.com/blargoner/math-algebra-greub-errata/blob/master/errata.pdf