Canonical Affine Structure of a Vector Space

up:: Vector Space

Let $(\vec{V},+,\cdot )$ be a vector space over a Field $\mathbb{K}$.

Then there is a canonical way of creating an Affine Space $(A,V,+)$ from it:

• Let the space’ set be the set $V$
• Let its underlying vector space be $\vec{V}$
• Let the Group Action be the additive group $(\vec{V},+)$

Thus, every vector space can be seen as an affine space over its own set $V$, acted upon by its own additive group $(\vec{V},+)$.

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References

• Basics of Affine Geometry