Vector Space

up:: 021a MOC Linear Algebra

A vector space over a Field (Algebra) $\mathbb{K}$ is a triple $(V,+,\cdot )$, where $V$ is a (non-empty) set, with operations $+$ and $\cdot$ under which

1. $(V,+)$ is an Abelian Group
   • The sum is commutative, associative, has identity $0$ and all elements have inverses
2. $\mathbb{K}$ induces a product by scalars upon $V$ (resembles a Group Action but from the field $\mathbb{K}$)
   • The product of scalars with vectors is commutative ($(\mu \nu )\cdot v=(\nu \mu )\cdot v$) and associative ($(\mu \nu )\cdot (\rho \cdot v)=\mu \cdot ((\nu \rho )\cdot v)$)
3. The sum and product operations are both distributive
   • $(\mu +\nu )\cdot v=\mu \cdot v+\nu \cdot v$
   • $\mu \cdot (u+v)=\mu \cdot u+\mu \cdot v$

Properties

Every vector space can be seen as an Affine Space, via the Canonical Affine Structure of a Vector Space.

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References

• abstract algebra - What is a “field action”, and how does it relate to group actions? - Mathematics Stack Exchange