# Hyperplane

• Nov 25th 2011, 07:34 AM
noricka
Hyperplane
In an n-dimensional space, how many vectors construct a hyperplane?

My guess is n, because a line in R^2 is a hyperplane and a plane in R^3 is a hyperplane. But how can I prove this?
• Nov 25th 2011, 08:16 AM
Deveno
Re: Hyperplane
n-1, not n.

a hyperplane is defined by a single equation of the form:

$a_1x_1 + a_2x_2 + \dots + a_nx_n = b$, where not all $a_i$ are 0.

the solution set consists of all elements of the solution space of the associated homogeneous system:

$a_1x_1 + a_2x_2 + \dots + a_nx_n = 0$, plus any (particular) vector $(b_1,b_2,\dots,b_n)$ lying on the hyperplane

(in other words we have a coset, or translate, of the nullspace of the homogeneous system).

since the homogeneous system has rank 1, its solution space has dimension n-1 (by the rank-nullity theorem), therefore....
• Nov 28th 2011, 04:20 AM
noricka
Re: Hyperplane
Thank you very much!
• Nov 28th 2011, 06:13 AM
HallsofIvy
Re: Hyperplane
What, exactly, do you mean by "construct a plane"? In n dimensional space, there will be, pretty much by definition, n-1 independent vectors lying on a hyperplane through the origin. But there will be no vectors lying on a hyper plane [b]not[/b ] through the origin. There is, however, always one vector perpendicular to the given hyperplane and that can be used to write the equation of the plane.

As Deveno said, any hyperplane can be written as $a_1(x_1- p_1)+ a_2(x_2- p_2)+ \cdot\cdot\cdot+ a_n(x_n- p_n)= 0$, where $$ is a vector normal to the hyperplane and $(p_1, p_2, \cdot\cdot\cdot, p_n)$ is a point in the hyperplane.