Vector Space with Standard Affine Structure is Affine Space
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Theorem
Let $E$ be a vector space.
Let $\struct {\EE, E, +, -}$ be the standard affine structure on $E$.
Then with this structure, $\EE$ is an affine space.
Proof
We are required to show that:
\((1)\) | $:$ | \(\ds \forall p, q \in \EE:\) | \(\ds p + \paren {q - p} = q \) | ||||||
\((2)\) | $:$ | \(\ds \forall p \in \EE: \forall u, v \in E:\) | \(\ds \paren {p + u} + v = p + \paren {u + v} \) | ||||||
\((3)\) | $:$ | \(\ds \forall p, q \in \EE: \forall u \in E:\) | \(\ds \paren {p - q} + u = \paren {p + u} - q \) |
By definition of the standard affine structure, the addition and subtraction operations are simply those in the vector space $E$.
That is, we want to show that:
\((1)\) | $:$ | \(\ds \forall u, v \in E:\) | \(\ds u + \paren {v - u} = v \) | ||||||
\((2)\) | $:$ | \(\ds \forall u, v, w \in E:\) | \(\ds \paren {u + v} + w = u + \paren {v + w} \) | ||||||
\((3)\) | $:$ | \(\ds \forall u, v, w \in E:\) | \(\ds \paren {v - u} + w = \paren {v + w} - u \) |
By definition the addition operation on a vector space is commutative and associative.
But all three axioms are immediate consequences of commutativity and associativity.
This concludes the proof.
$\blacksquare$
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