Hyperplanes

   A hyperspace in    consists of all vectors which are perpendicular to a given non-zero vector. Thus a hyperspace always contains the origin. If we (parallel-)translate it away from the origin, we obtain a hyperplane in  .

Here you can learn how to describe hyperplanes mathematically. Later this will help you interpret the solutions of a system of linear equations geometrically. - We begin with a definition of hyperspace.

Definition   Given a nonzero vector  n in  , the hyperspace perpendicular to  n  is the set
 

The vector n is called  a normal vector of  Perp(n).

   We will express the dot product equation  x n 0  by an equivalent equation in variables. It is very useful to be become thoroughly familiar with these two points of view, as the dot product version lends itself to a conceptual and geometrical interpretation of such equations. Whereas the -variables point of view readily extends itself to systems of linear equations which we will study in detail lateron. We use coordinates

x ( , ... , )    and    n ( , ... , )

then  Perp(n) consists of the solutions of the equation in variables  , ... ,

+ ... + 0.

Conversely, given an equation

+ ... + 0,    with   n ( , ... , ) 0,

we know that its solutions form the hyperspace  Perp(n).

Examples   of hyperspaces in 2-space and in 3-space.

   Now let us turn to hyperplanes. Visual inspection suggests: The location of a hyperplane    is completely determined by a vector  n  which is perpendicular to  , and a point    contained in  . Accordingly, we have

Proposition   Given a nonzero vector  n  in    and a point    with position vector  p, the hyperplane perpendicular to  n  through    is the set of  all those  x  in    with
 
(x – p) n 0.    

The vector n is called a normal vector of the specified hyperplane.

   We emphasize that the dot product equation  (x – p) n 0  is equivalent to either one of the equations below:

x n     and     + ... + k,

where  + ... + , and

x ( , ... , ),    n ( , ... , ),    p ( , ... , )

Therefore, given an equation in variables of the form

+ ... +

in which is a constant and at least one  , we know that its solutions form the hyperplane which is perpendicular to  n ( , ... , )  and which passes through    with position vector

q (0 , ... , / , ... , 0),   where  0.

   Examples   of hyperplanes in 2-space and in 3-space

   Exercises   on hyperspaces and hyperplanes