Is the notion of "affineness" more general than "linearity", or vice versa? 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.
 A: 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.
A: 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.
