In a undergraduate text for linear algebra, the author writes a function
$F: \boldsymbol{V} \rightarrow \mathbb{R}^{2}$
where V is a vector space, and the author doesn't provide a mathematical formulation of this function. The chapter in question is about the isomorphism beetween vectors and arrays of real numbers.
My question is : since $\mathbb{R}^{2}$ is a vector field itself, what's the meaning in defining a function $\boldsymbol{V} \rightarrow \mathbb{R}^{2}$ ???
The meaning is that $V$ is any vector space (it can be $\mathbb R^k$ for some $k$, a vector space of polyomials or functions, or matrices, or whatever) and you are mapping that vector space into the specific vector space $\mathbb R^2$.
$F$ need not be any linear or simple function, it simply takes an element of $V$ and outputs an element of $\mathbb R^2$. For instance, if $V$ is the set of polynomials with real coefficients of degree 2, then for instance $F$ can be such that $$F:a+bX\mapsto(a+b,a-b),$$ which is actually an isomorphism.