# The Unapologetic Mathematician

## Affine Spaces

Today we’re still considering the solution set of an inhomogenous linear system $a_i^jx^i=y^j$ and its associated homogenous system $a_i^jx^i=0$. Remember that we also wrote these systems in more abstract notation as $Ax=y$ and $Ax=0$. The solution space to the homogenous system is the kernel $\mathrm{Ker}(A)$, and any two solutions of the inhomogenous system differ by a vector in this subspace.

We call such a collection of points an “affine space”. We can also talk about such a thing from the inside, without seeing it as embedded in some ambient vector space as a coset. Algebraically, we characterize an affine space $S$ as a collection of points and a function $\Theta:S\times S\rightarrow V$, where $V$ is some associated vector space. The value $\Theta(a,b)$ — also written $a-b$ — is thought of as the “displacement vector” from $b$ to $a$.

We require two properties for this map: first that $\Theta(a,b)+\Theta(b,c)=\Theta(a,c)$; second that for every $b$ the map $a\mapsto\Theta(a,b)$ from $S$ to $V$ is a bijection. The former condition provides coherence for the interpretation as displacement vectors. The latter implements the idea that an affine space $S$ “looks like” the associated vector space $V$.

Given a subspace $V$ of a vector space $W$, any coset $w+V$ is an affine space associated to $V$. As a degenerate case, we can consider $V$ as a subspace of itself, and $V$ itself is its only coset. Thus any vector space can be considered as an affine space associated to itself. In fact, since any affine space is in bijection with its associated vector space, we can get any one of them by this construction. Thus any two affine spaces associated to a given vector space are isomorphic, but not canonically so. It’s this lack of a canonical isomorphism that makes things interesting, because we can’t justify simply identifying non-canonically isomorphic spaces.

Another consequence of the bijection is that we can “add” a vector $v$ to a point $b$ in an affine space. Since $a\mapsto\Theta(a,b)$ is a bijection, there must be a unique point we’ll call $b+v$ so that $\Theta(b+v,b)=v$. It’s straightforward from here to show that this gives an action of the vector space $V$ (considered as an abelian group) on the affine space $S$.

In fact, considering an affine space in the category of $V$-actions, the bijection shows us that $S$ is isomorphic to $V$‘s action on itself by addition. We can even use this to characterize an affine space exactly as a $V$-set isomorphic to $V$‘s action on itself. In other words, it’s a torsor for $V$.

July 16, 2008 - Posted by | Algebra, Linear Algebra

1. Totally OT, but I was wondering how many years it takes to produce a competent professional mathematician. I was thinking about 12:

c. 2 yrs focussed work in secondary school (AP classes,
after-hours study guided by teacher, relative or
somebody)
2 years postdoc.

Which puts it into roughly the ballpark of acquiring adult-like competence at speaking a language

Comment by Avery Andrews | July 17, 2008 | Reply

2. Well, actually most of my fellow graduate students where I did my Ph.D. just sort of fell into it after college. Yeah, they were smart and such in high school, and yeah they majored in math or something close. But they didn’t focus on becoming mathematicians until they decided they didn’t want to get a real job after graduating.

Myself, I’ve been aiming at this since I became aware it was an option. So I was something like 11 or 12 at the time. And I wouldn’t say I’ve “made it” yet, given that I keep just barely finding jobs by the skin of my teeth.

Comment by John Armstrong | July 17, 2008 | Reply

3. How long until Mastery of the Glass Bead Game?

In September 2007 I made a comment on the n-Category Cafe that included my hypothesis that: “there seems to be a rule of thumb that it takes an adult human being on the order of 10,000 hours of study to reach professional competence in almost any subject, from Music to Mathematics. It is no coincidence that this is how long a full-time student spends in a given university.”

“Counterexamples have been made to my comments on the likes of Mozart (who appears to have been composing while still in utero), Gauss, Ramanujan, Terry Tao, Feynman, and Newton. Counter-counter comments were made. I’m not sure what conclusion, if any, was reached, but I still hold (perhaps more loosely) the 10,000 hour study rule.”

“For children, the line between Play and Study is fuzzy at best. My strength and weakness as a scholar is that I still feel that I’m playing with Math, Physics, Computing, Biology, and Literature…”

Comment by Jonathan Vos Post | July 17, 2008 | Reply

4. […] the card game I’d like to talk about the card game Set. Why? Because the goal is to find affine lines! […]

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5. […] homogenous system . If there are any solutions of the system under consideration, they will form an affine space of this […]

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6. […] space that still make sense in the absence of an origin. Indeed, we can still talk about it as an affine space, where the idea of displacement vectors between points still makes sense. And these displacement […]

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