Exercise 4.6 of An algebraic introduction to mathematical logic asks: $K$ is a field. Show that vector spaces over $K$ form a variety $V$ of algebras, and that every space over $K$ is a free algebra of $V$.
They give the following definition for free algebra:
a T-algebra $R$ in the variety $V$ is the free algebra of $V$ on the set $X$ of free generators (where a function $\sigma: X\to R$ is given, usually as an inclusion) if, for every algebra $A$ in $V$ and every function $t: X\to A$ there exists a unique homomorphism $p: R\to A$ such that $p\sigma = t$.
Given that definition, it doesn't seem to me that $K$ can be a free algebra of $V$, it has to be a free algebra of $V$ on some set $X$. Is that correct? If so, what set am I implicitly supposed to assume it is a free algebra on?
Pick a basis $X$ for your vector space $R$. (For infinite-dimensional spaces you need the axiom of choice.) Here $\sigma:X \to R$ is the inclusion map. Then every function from $X$ to another vector space extends uniquely to a linear map.