- Definition (Vector Space)
- Let F be a field. A set V with two binary operations: + (addition) and
(scalar multiplication), is called a Vector Space if it has the following properties:
forms an abelian group
for
and ![{\displaystyle v\in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb)
for
and ![{\displaystyle v,u\in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feb5317ad7d3af5b08f8f324185632bba5088ae3)
![{\displaystyle (ab)v=a(bv)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66d0e29bf06f635175bad019ca2f1eb8ccde69c2)
![{\displaystyle 1_{F}v=v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/397fc293cbaad0a170587622b8a5bcffe8188bf9)
The scalar multiplication is formally defined by
, where
.
Elements in F are called scalars, while elements in V are called vectors.
- Some Properties of Vector Spaces
![{\displaystyle 0_{F}v=0_{V}=a0_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982dbf1f3e6e2c02ac10e9dc90666ab27bada71b)
![{\displaystyle (-1_{F})v=-v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/168d7b53098bb217b6ed3b29c25a9a27e3bb0977)
![{\displaystyle av=0\iff a=0{\text{ or }}v=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b07877534c46a4ab00d5e6336ce312faaf29ceb)
- Proofs:
![{\displaystyle 0_{F}v=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a(0_{V}+0_{V})=a0_{V}+a0_{V}\Rightarrow a0_{V}=0_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b879e3750d76dde68712a26bcb1e40d27342ecb)
- We want to show that
, but ![{\displaystyle v+(-1_{F})v=1_{F}v+(-1_{F})v=(1_{F}+(-1_{F}))v=0_{F}v=0_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b11fc159d827172c68e444b7f3e6e7ea66479d9d)
- Suppose
such that
, then ![{\displaystyle a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7c954813f9f9c9e19df21cc4fb59e1e02fbe86)