In combinatorics, a (d-1)-dimensional flat in R^d.

back to combinatorics--

artemis entreri for non-combinatorists...
A hyperplane is a generalization to V=Rd (or even to V=Fd, for some more exotic field F) of the concept of a plane in the (somewhat) familiar "euclidian 3-space" R3. 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 Rd (or Fd, if using). This is nicely (d-1)-dimensional, but when d=3 it still fails to capture the concept of a plane in R3: 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 R3 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∈Rd.

Other equivalent formulations for Rd abound; just remember: a hyperplane is really little more than a plane in higher dimensions!

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