I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?

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    $\begingroup$ It's the surface of your favorite table inside your math lounge :) $\endgroup$ – Hui Yu Feb 1 '13 at 15:16

Think of a line in the plane. If you shift that line parallel to itself, by all possible amounts, you fill up the whole plane.

Now, think of a plane in 3-dimensional space. If you shift that plane parallel to itself, by all possible amounts, you fill up all of 3-space.

A hyperplane is something you can shift parallel to itself, by all possible amounts, and by so doing, fill up all of space.

  • $\begingroup$ thank you for your explanations! $\endgroup$ – Stephane Kouakou Feb 1 '13 at 15:44
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    $\begingroup$ Of course, all the lines parallel to a given line also fill 3-space; the shifts have to all be in one chosen direction for this to work. Also, one generally excludes curvy sets (like $y=\cos(x)$ in the plane) which nonetheless fill up space with their translations. $\endgroup$ – Kundor Sep 3 '14 at 19:44

Hyperplane, in finite dimensional linear algebra (or geometry) is a subspace (or a translation of a subspace) of dimension one less than the whole space's.

Thus, in the plane $\Bbb R^2\,$ , any line is a hyperplane (that must pass through the origin if we require it to be a subspace), in the space $\,\Bbb R^3\,$ any plane is a hyperplane, etc.

In general, and in any dimension, in a linear (vectorial) space $\,X\,$ a subspace $\,H\,$ is a hyperplane iff it has codimension $\,1\,$ iff $\,H=\ker\phi\,$ , for some $\,0\neq \phi\in X^*\,$ iff it is a proper subspace of maximal dimension in $\,X\,$ , meaning:

$$\forall\,x\in X-U\,\,,\,\,Span\{U,x\}=X$$


Affine subspaces of the 3d space ($\Bbb R^3$) are all points ($0$ dim. subspaces), all lines ($1$ dim. subspaces), all planes ($2$ dim. subspaces) and the unique $3$ dim. subspace, the whole space.

Those which contain the origo, are called (linear) subspaces.

At a corner of a room in the $n$ dimensional space, there are not $3$ but $n$ segments, pairwise orthogonal to each other, meeting in the corner point.

A hyperplane -- within an $n$ dim. space -- is an $n-1$ dimensional subspace, also called $1$-codimensional subspace.

  • $\begingroup$ thank you for your explanations! $\endgroup$ – Stephane Kouakou Feb 1 '13 at 15:47

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