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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?

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$K$ is a vector space over $F$, and you want to verify that the map $f\colon K \to K$ defined by $f(x) = \alpha x$ is an $F$-linear transformation in the sense of linear algebra. In other words, for $x, y \in K$ you want $f(x + y) = f(x) + f(y)$ and for $a \in F$, $f(ax) = af(x)$. – Dylan Moreland Feb 5 '12 at 21:02
You can see $K$ as a vector field over $F$. $F$-linear is linear transformation of $K$ as a vector field over $F$. – azarel Feb 5 '12 at 21:03
up vote 3 down vote accepted

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.

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