I'm having trouble answering the last problem Linear Algebra set. Not looking for a solution, of course, but some pointers would be incredibly helpful.

Given a vector space $F^S$ of all functions from the set $S$ to the field $F$, how do you show that $F^S$ is finite dimensional if and only if the set $S$ is finite?

I'd generally attempt to construct a basis for $F^S$, but I can't wrap my head around creating a list of functions that span $F^S$.


This may or may not be a helpful suggestion.

Can you consider $\mathbb{F}^n$, the $n$ dimensional vector space over $\mathbb{F}$, as a vector space of the given form? That is, can you choose a set $S$ such that $\mathbb{F}^S$ and $\mathbb{F}^n$ are isomorphic? If so, then if you can pick out a basis for $\mathbb{F}^n$, you should be able to use the isomorphism to help you determine how to choose a basis for vector spaces of the form $\mathbb{F}^S$.

  • $\begingroup$ That's actually super helpful, thanks! $\endgroup$ – John Pellton Oct 1 '12 at 6:44

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