Vector Space

up:: 021a MOC Linear Algebra

A vector space over a Field (Algebra) is a triple , where is a (non-empty) set, with operations and under which

1. is an Abelian Group
   • The sum is commutative, associative, has identity and all elements have inverses
2. induces a product by scalars upon (resembles a Group Action but from the field )
   • The product of scalars with vectors is commutative () and associative ()
3. The sum and product operations are both distributive

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