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This is an incredibly dumb question, so please bear with me.

An affine transformation $T$ is equal to a linear transformation $L$ plus a translation $t$. This suggests that affine transformations are more general than linear transformations, because for the former $t$ can be non-zero, but for the latter $t$ must be zero.

Likewise, an affine subspace is of the form $X+c$, where $X$ is a linear subspace. Since $c=0$ necessarily for a linear subspace, but not for an affine one, this suggests that affine subspaces are more general and that linear subspaces are a special case.

However, an affine combination is a linear combination, with the additional restriction that the sum of the coefficients has to equal $1$. This suggests that affine combinations are special cases of linear combinations, and that the latter is more general.

Question: Which notion is more general, "affineness" or "linearity"? Affine transformations and affine subspaces suggests that "affineness" is more general, but affine combinations suggest that "linearity" is more general. Why do these concepts suggest opposite conclusions?

EDIT: An alternative definition for affine subspace is a set which is closed under affine combinations, so this suggests to me some sort of underlying duality, although I am not sure.

Chill2Macht
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    This is not a dumb question at all. On the contrary, the kind of "duality" you have noticed here is insightful (which is compactly summarized in Oscar's answer below). This idea has never occurred to me personally, so I am glad to have learnt it now. Thanks for writing the post! – Prism Sep 08 '16 at 22:28
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    @Prism Thank you for the feedback! Another way to think about Oscar's answer which occurred to me recently is that it is the same idea as the proof of why $X \subset Y \implies X^{\perp} \supset Y^{\perp}$, where $\perp$ denotes the orthogonal complement. – Chill2Macht Sep 09 '16 at 00:47
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    Yes! It is also similar to the following: If $X\subset Y$ are two subsets of the affine space $\mathbb{A}^{n}$, then $I(X)\supset I(Y)$. Here, I'm using the notation $I(Z)={f\in k[x_1, ..., x_n]: f(z)=0 \text{ for all } z\in Z}$ for a given subset $Z\subset\mathbb{A}^n$. – Prism Sep 09 '16 at 07:28
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    Galois connection – Oscar Cunningham Sep 09 '16 at 08:37
  • Another similar example for future reference of anyone seeing this post for the first time: finer topologies, which have more open sets and thus more possible open covers, have fewer compact sets, which have to have a finite subcover for every possible open cover, while coarser topologies, which have fewer open sets and thus fewer open covers, have more compact sets. – Chill2Macht Sep 11 '16 at 06:56
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    Any affine space can be turned into a linear space by an arbitrary choice of one point (the origin). Any vector space can be considered an affine space where the set of points and the set of translation vectors coincide. So they're not the same concept, but they're convertible concepts -- if that makes any sense. –  Sep 15 '16 at 03:21

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There is often an informal duality between generality in your definitions and lack of generality in your operations. If you make a definition more lax, allowing more structures that satisfy the definition (i.e., make them more general), then your constructions typically need to be restricted to a smaller class, because they need to work for all the new structures as well.

For instance, "groups" are more general than "abelian groups". However, when talking about groups, a normal subgroup is a subgroup with an additional condition; for abelian groups, a normal subgroup is the same as a subgroup.

A similar thing happens in your example. Expanding the definition of "subspace" to include affine subspaces and not just linear subspaces, restricts the "combination" operation, whose purpose it is to make new vectors in the space from other vectors in the space, because it has to work even on subspaces which are "merely" affine, not linear.

Mees de Vries
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  • I really like the group theory example. It's especially pertinent too given that affine subspaces are the cosets of linear subspaces (under addition). – Chill2Macht Sep 07 '16 at 17:57
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Yeah you're right about the duality

Affine transformations have to preserve affine combinations.
Linear transformations have to preserve linear combinations.

and

Affine subspaces have to be closed under affine combinations.
Linear subspaces have to be closed under linear combinations.

So since there are more linear combinations, there are fewer linear transformations and subspaces.

Oscar Cunningham
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