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 | 6 Comments