Vector Space

Topics: Algebra

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(definition)

Let $K$ be a field and $V$ a set equipped with two operations:

$$+:V\times V\to V$$

$$\cdot :K\times V\to V$$

We say that $V$ is a $K$-vector space if the properties below are satisfied.

An example of a vector space is the set of vectors in ${\mathbb{R}}^2$.

Properties §

$\forall u,v,w\in V$, $\forall s,t\in K$

1. $u+(v+w)=(u+v)+w$, $\forall s$
2. $u+v=v+u$
3. $\exists 0\in V$ such that $u+0=u$
4. $\forall u\in V,$ $\exists (-u)$ such that $u+(-u)=0$
5. $t\cdot (s\cdot w)=(ts)\cdot w$
6. $1\cdot u=u$, $1\in K$
7. $s\cdot (u+v)=(s\cdot u)+(s\cdot v)$
8. $(s+t)\cdot u=(s\cdot u)+(t\cdot u)$