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:

$\begin{array}{lr} f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$

Abelian groups