Fixing a point in an affine space induces a vector space

up:: Affine Space

Let $(A,V,+)$ be an affine space, and let $O\in A$ be a fixed point (which will play the role of the origin). For all $x\in A$, write $\overrightarrow{Ox}=O-x\in V$ as the vector separating $O$ and $x$.

Then the function

$$\begin{array}{rl}{m}_O:& A\to V\\ & x\to \overrightarrow{Ox}\end{array}$$

is a Bijection¹, with inverse

$$\begin{array}{rl}{m}_O^{-1}:& V\to A\\ & v\to O+v\end{array}$$

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

$$\begin{array}{rl}x+y& :=m_O^{-1}(\overrightarrow{Ox}+\overrightarrow{Oy})\\ \lambda x& :=m_O^{-1}(\lambda \overrightarrow{Ox})\end{array}$$

Thus, $A$ can be seen as a vector space, with the fixed point $O$ 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 $A$ there is only one $v\in V$ ─ thus, being trivially isomorphic to the entire vector space $V$.

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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 $V$, as well as the regularity of the action (that is, there are no “lazy vectors” allowed; all vectors “induce movement” upon the space). ↩