I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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.
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:
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.