Fixing a point in an affine space induces a vector space

up:: Affine Space

Let be an affine space, and let be a fixed point (which will play the role of the origin). For all , write as the vector separating and .

Then the function

is a Bijection¹, with inverse

Using these operators, one can create the Vector Space operations on as

Thus, can be seen as a vector space, with the fixed point acting as its origin.

When seeing the affine space as a vector space, An affine space with a fixed point is isomorphic to its underlying vector space², since to each point in there is only one ─ thus, being trivially isomorphic to the entire vector space .

---

References

• Affine transformation - Wikipedia

Footnotes

1. It is Surjective due to the space’s Group Action being Transitive, and Injective due to the action being Free ─ i.e. due to the group action being Regular. ↩

2. This is true if the entire vector space spans the affine space, which we pressupose due to the group action’s domain covering the entirety of , as well as the regularity of the action (that is, there are no “lazy vectors” allowed; all vectors “induce movement” upon the space). ↩