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### Affinity in Generality

Let *V* be a vector space. Consider the set of all affine transformations of *V:* An *affine transformation* of *V* is a map from *V* to itself which can be expressed as

,

for some invertible linear map (automorphism) *L* and some vector *b* in *V*. The set of all affine transformations of *V* forms a group under composition and is called the *affine group* of *V*. Note that the set of automorphisms of *V* is a subgroup of the affine group and also that the set of translations of *V* is also subgroup of the affine group. Note that every element of the affine group can be expressed as a composition of an invertible linear map followed by a translation. Note also that the only affine transformation which is both an automorphism and a translation is the identity map. It is also quite easy to see that the translation group is a normal subgroup of the affine group. However, the automorphism group of *V*, which, since *V* is a vector space is known as the *general linear group* of *V*, is not.

Is the affine group the internal direct product of the translation group and the automorphism group? No, but it comes close. In the previous post, we saw that if *G* is the internal direct product of *H* and *K, *then not only can every element of *G* can be expressed (uniquely) in the form *g = hk,* but also the elements of *H* commute with the elements of *K*. This is the sense in which the two groups *H* and *K do not interact* with each other inside *G*. In the case of the affine group, there is interaction: The translation group of a vector space does not commute with its general linear group. This breakdown is evidenced by the fact that one of the two groups fails to be normal.

Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.

Suppose I have two groups *H* and *K*. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product *H**K, *and bestows it with a group operation in the most basic and obvious way possible: *(h,k)**(h’,k’) = (hh’,kk’). *The group that results is known as the *direct product* of *H* and *K.*

The same can be done with vector spaces: Given two vector spaces *V* and *W*, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form *(v,w) *with *v* in *V* and *w* in *W*. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space *X *and two subspaces *Y* and *Z* such that every vector *x* in *X* can be written uniquely in the form *x = y + z, *for *y* in *Y* and *z* in *Z*. I would then be invited to observe how this means that *X* was the direct sum of *Y* and *Z.* This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = *y + z* was the same as x = *(y,z)*.