The following proposition is in Benson's book:
Proposition 4.1.2 Suppose $k$ is algebraically closed. Let $A_1,A_2\in M_{n\times m}(k)$, regarded as maps from $k^m$ to $k^n$. Then: (1) Suppose that forall $\lambda,\mu\in k$, not both zero, $\lambda A_1+\mu A_2$ is injective. Then $$\bigcap_{(\lambda,\mu)\neq (0,0)}Im(\lambda A_1+\mu A_2)=0$$ (2) Suppose that forall $\lambda,\mu\in k$, not both zero, $\lambda A_1+\mu A_2$ is surjective. Then $$\sum_{(\lambda,\mu)\neq (0,0)}Ker(\lambda A_1+\mu A_2)=k^m$$
This is very interesting proposition and the proof in the book is very beautiful. The proof in the book used field extension. I am wondering if this can be proved by methods of linear algebra. Any helps will be appreciated.
The following is proof in the book:
Th4.1.1 is that the rank is preserved when consider field extension and take algebraic independent variables.