artemis entreri for non-combinatorists...

A hyperplane is a generalization to V=**R**^{d} (or even to V=F^{d}, for some more exotic field F) of the concept of a plane in the (somewhat) familiar "euclidian 3-space" **R**^{3}. Formally, it is a linear manifold of dimension `d-1`.

A "linear manifold" is really very simple: Take some `(d-1)`-dimensional vector subspace W of the `d`-dimensional vector space **R**^{d} (or F^{d}, if using). This is nicely `(d-1)`-dimensional, but when `d`=3 it still fails to capture the concept of a plane in **R**^{3}: of necessity, 0∈W, whereas real planes don't necessarily go through the origin!
We "fix" this by picking some vector v which we want to be in the hyperplane, and considering the sum L=v+W. Every plane in **R**^{3} may be expressed in this form (and only planes are expressed thus), so this is a generalization; it turn out also to be useful...

Note that if v'-v∈W, then v+W=v'+W -- there is more than one choice of the vector v for the hyperplane. However, it can be seen that the choice of W is unique -- no other `(d-1)`-supspace gives the same hyperplane.

When doing geometry in euclidian space, we also have an inner product and a notion of orthogonality. This is useful in giving a more convenient formulation of hyperplanes. Instead of picking an arbitrary vector v, we pick the *shortest* vector v connecting the origin to our hyperplane. This vector is unique, and is orthogonal to W: v⊥W. It turns out that

W = {u: v⊥u}.

So any hyperplane

*not passing through the origin* is uniquely determined by a single vector 0≠v∈

**R**^{d}.

Other equivalent formulations for **R**^{d} abound; just remember: a hyperplane is really little more than a plane in higher dimensions!