Let $F$ be the set of all complex-valued functions $f:X \to \mathbb{C}$ defined on the set $X =${1, 2, 3} with 3 elements. Show that $F$ is a vector space over the field $\mathbb{R}$ of real numbers. Describe an isomorphism between $F$ and the real vector space $\mathbb{R}^n$, for some integers to be determined. Is this isomorphism unique?
I tried to prove $F$ is a vector space by checking 8 conditions. And then I do not know how to construct a isomorphism between $F$ and the real vector space $\mathbb{R}^n.$
Note that defining a function $f:X\to \mathbb C$ is the same as defining the values of $f(1)$, $f(2)$ and $f(3)$, that is, three complex numbers. So the obvious mapping here is
$\phi: F\to \mathbb C^3, \phi(f)=(f(1), f(2), f(3))$
But we're working over the field $\mathbb R$, so we'd like to write $\mathbb C^3$ in the form $\mathbb R^n$ for a certain natural $n$. Now, recall that any complex number $z=x+iy$ can be written as a pair $(x,y)$. So we now can rewrite the mapping above like this:
$\phi: F\to \mathbb R^6, \phi(f)=(Re\ f(1), Im\ f(1), Re\ f(2), Im\ f(2),Re\ f(3), Im\ f(3))$
So, to finish, you just have to show that $\phi$ is an isomorphism.