Intuition about Hyperplane I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
 A: 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. 
A: 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:
$$\forall\,x\in X-U\,\,,\,\,Span\{U,x\}=X$$
A: 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.
