TITLE: What does it mean for an action to be an $F$-linear transformation?
QUESTION [6 upvotes]: I'm working on a problem from Dummit & Foote's Abstract Algebra and I can't figure out what exactly I'm being asked to prove. I hate to ask this here, because it seems that I should've been able to find an answer on my own. The problem is from Section 13.2 #19 and it reads:

Let $K$ be an extension of $F$ of degree $n$.
(a) For any $\alpha\in K$ prove that $\alpha$ acting by left multiplication on $K$     is an $F$-linear transformation of $K$.

I have searched and searched through the text and around on the internet and I cannot find an explicit definition of an "$F$-linear transformation".  What does this mean?
If there is a definition for this in the book (or on the net) could anyone direct me to it?

REPLY [7 votes]: Since $F$ is a subfield of $K$, there is a natural $F$-vector space structure on $K$. Being $F$-linear just means that it is a linear transformation on $K$ where you are viewing $K$ as this $F$-vector space.