Thm. Let $L$ be a solvable subalgebra of $gl(V)$, $V$ a finite dimensional nonzero vector space. Then $V$ contains a common eigenvector for all the endomorphisms in $L$.
The proof of this theorem is typically broken down into 4 parts, 1) find an ideal $K$ of codimension one, 2) show that common eigenvectors exist for $K$, 3) verify that $L$ stabilizes a space consisting of such eigenvectors, 4) find an eigenvector for a single $z\in L$ satisfying $L=K+Fz$.
For part 2, Humphreys says(p.16), there exists a linear functional $\lambda: K\rightarrow F$ satisfying $x.v=\lambda (x)v$, $x\in K$.
This may be trivial, but I do not recall this fact from basic linear algebra. I would like to understand this and/or get a reference for this statement. Thanks.