PlanetPhysics/Vector Space 2

Let $F$ be a [[../CosmologicalConstant/|field]] (or, more generally, a division ring). A vector space $V$ over $F$ is a set with two [[../Cod/|operations]], $+:V\times V⟶V$ and $\cdot :F\times V⟶V$, such that

1. $(𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰)$ for all $𝐮,𝐯,𝐰\in V$
2. $𝐮+𝐯=𝐯+𝐮$ for all $𝐮,𝐯\in V$
3. There exists an element $\mathrm{𝟎}\in V$ such that $𝐮+\mathrm{𝟎}=𝐮$ for all $𝐮\in V$
4. For any $𝐮\in V$, there exists an element $𝐯\in V$ such that $𝐮+𝐯=\mathrm{𝟎}$
5. $a\cdot (b\cdot 𝐮)=(a\cdot b)\cdot 𝐮$ for all $a,b\in F$ and $𝐮\in V$
6. $1\cdot 𝐮=𝐮$ for all $𝐮\in V$
7. $a\cdot (𝐮+𝐯)=(a\cdot 𝐮)+(a\cdot 𝐯)$ for all $a\in F$ and $𝐮,𝐯\in V$
8. $(a+b)\cdot 𝐮=(a\cdot 𝐮)+(b\cdot 𝐮)$ for all $a,b\in F$ and $𝐮\in V$

Equivalently, a vector space is a [[../RModule/|module]] $V$ over a ring $F$ which is a field (or, more generally, a division ring).

The elements of $V$ are called [[../Vectors/|vectors]], and the element $\mathrm{𝟎}\in V$ is called the zero vector of $V$.

This entry is a copy of the GNU FDL vector space article from PlanetMath. Author of the original article: djao. History page of the original is here