How would I go about determining the basis for the vector space $F(X,V)$ of all functions mapping an element of any arbitrary set $X$ to some vector space $V$ with the usual definition of addition and scalar multiplication on functions, should I already know the basis for that vector space?
I tried to extend the notion of the basis of the vector space of linear maps $F(X,F)$ from some set $X=\{a_0,...,a_n\}$ to an arbitrary field where the basis was made up of functions defined as $f_i(a_j)=1$ when $i=j$ and $0$ otherwise. However I have had no such luck developing the idea for $F(X,V)$.
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $x\in X$ and each $w\in B$, let $f_{x,w}\in F(X,V)$ be the function defined by$$f_{x,w}(y)=\begin{cases}w&\text{ if }y=x\\0&\text{ otherwise.}\end{cases}$$Then $\{f_{x,w}\,|\,x\in X\wedge w\in B\}$ is a basis of $F(X,V)$.