Abbreviation: FVec

A vector space over a field $\mathbf{F}$ 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 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 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