Abbreviation: FVec
A \emph{vector space over a fields} $\mathbf{F}$ is a structure $\mathbf{V}=\langle V,+,-,0,f_a\ (a\in F)\rangle$ such that
$\langle V,+,-,0\rangle $ is an abelian groups
scalar product $f_a$ distributes over vector addition: $a(x+y)=ax+ay$
$f_{1}$ is the identity map: $1x=x$
scalar product distributes over scalar addition: $(a+b)x=ax+bx$
scalar product associates: $(a\cdot b)x=a(bx)$
Remark: $f_a(x)=ax$ is called \emph{scalar multiplication by $a$}.
Let $\mathbf{V}$ and $\mathbf{W}$ be vector spaces over a field $\mathbf{F}$. A morphism from $\mathbf{V}$ to $\mathbf{W}$ is a function $h:V\rightarrow W$ that is \emph{linear}:
$h(x+y)=h(x)+h(y)$, $h(ax)=ah(x)$ for all $a\in F$
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$