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Given two groups $G$ and $H$, it is natural to declare that they have the same structure if there is a bijective map $f:G\to H$ such that $f(g_1g_2)=f(g_1)f(g_2)$ for all $g_1$ and $g_2\in G$. This says that the members of $H$ 'multiply the same way' as the corresponding members of $G$ under this mapping $f$.

This motivates the definition of a group homomorphism.

I recently read about affine spaces and the definition of an affine transformation. I cannot quite see how an affine isomorphism preserves the affine structure. Here is what I know.

Definiton. An affine space $P$ is a set along with a vector space $V$ acting on it transitively and freely.

Definition. Let $F$ be a field and $P$ be an affine space over the vector space $(V,F)$. Let $p_1,\ldots, p_n$ be points in $P$. Then for $a_1,\ldots, a_n\in F$ satisfying $a_1+\cdots +a_n=1$, we define $$p_1\cdot a_1+\cdots +p_n\cdot a_n = a_1(p_1-p)+\cdots +a_n(p_n-p)+p$$ where $p$ is a point in $P$.

It can be checked that this definition is unambiguous, i.e., it doesn't depend on the choice of the point $p$.

Definition. Let $F$ be a field. Let $P$ be an affine space over $(V,F)$ and $Q$ be an affine space over $(W,F)$. An affine transformation from $P$ to $Q$ is a mapping $f:P\to Q$ such that $$f(p_1\cdot a_1+\cdots p_n\cdot a_n)=f(p_1)\cdot a_1+\cdots +f(p_n)\cdot a_n$$

for all $p_1,\ldots, p_n\in P$ and $a_1,\ldots, a_n\in F$ satisfying $a_1+\cdots +a_n=1$.

Definition. An affine isomorphism is a bijective affine transformation.

I cannot see how an affine isomorphism, as defined above, captures the notion of structure preservation. The algebraic structure of an affine space is also making me a bit nauseous. It is a set acted upon by a vector space in a special way, where the latter itself has an algebraic structure.

Can somebody please enlighten me on how to understand an affine isomorphism as an affine structure preserving map?

Thank you so much for reading this long post.

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    $\begingroup$ The affine stuff is roughly the linear stuff plus a displacement of the origin. I am not sure what the simpler case of linear transformations ought to preserve. Perhaps mapping linear spaces to linear spacese? $\endgroup$ – mvw May 9 '15 at 22:07
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    $\begingroup$ In case of Linear isomorphisms, I think the definition is a natural one. The structure on a linear space is governed by the vector aaddition and scalar multiplication. If $T:V\to W$ is a linear isomorphism, then it's bijective and preserves additive structure: $T(u+v)=T(u)+t(v)$, and multiplicative structure $T(\alpha u)=\alpha T(u)$. $\endgroup$ – caffeinemachine May 9 '15 at 22:29

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