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Hyperplane, Subspace and Halfspace

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1. Hyperplane : Geometrically, a hyperplane is a geometric entity whose dimension is one less than that of its ambient space.


What does it mean? 
It means the following. For example, if you take the 3D space then hyperplane is a geometric entity that is 1 dimensionless. So it’s going to be 2 dimensions and a 2-dimensional entity in a 3D space would be a plane. Now if you take 2 dimensions, then 1 dimensionless would be a single-dimensional geometric entity, which would be a line and so on. 

  • The hyperplane is usually described by an equation as follows 

    XT n + b =0

  • If we expand this out for n variables we will get something like this 

    X1n1 + X2n2 + X3n3 + ……….. + Xnnn + b = 0

  • In just two dimensions we will get something like this which is nothing but an equation of a line. 

    X1n1 + X2n2 + b = 0

Example: 

Let us consider a 2D geometry with
 n = \begin{bmatrix} 1\\ 3\\ \end{bmatrix} and\ b = 4 
Though it's a 2D geometry the value of X will be
 X = \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix} 
So according to the equation of hyperplane it can be solved as
 X^Tn + b = 0\\ \begin{bmatrix} x_1 x_2 \end{bmatrix} \begin{bmatrix} 1\\ 3\\ \end{bmatrix} + 4 = 0\\ x_1 + 3x_2 + 4 = 0  
So as you can see from the solution the hyperplane is the equation of a line. 
2. Subspace : Hyper-planes, in general, are not sub-spaces. However, if we have hyper-planes of the form,

XT n =0

That is if the plane goes through the origin, then a hyperplane also becomes a subspace. 3. Half-space : Consider this 2-dimensional picture given below.


So, here we have a 2-dimensional space in X1 and X2 and as we have discussed before, an equation in two dimensions would be a line which would be a hyperplane. So, the equation to the line is written as

XT n + b =0 

So, for this two dimensions, we could write this line as we discussed previously 

X1n1 + X2n2 + b = 0

You can notice from the above graph that this whole two-dimensional space is broken into two spaces; One on this side(+ve half of plane) of a line and the other one on this side(-ve half of the plane) of a line. Now, these two spaces are called as half-spaces. Example:

Let’s consider the same example that we have taken in hyperplane case. So by solving, we got the equation as 

x1 + 3x2 + 4 = 0

There may arise 3 cases. Let’s discuss each case with an example.

Case 1:

x1 + 3x2 + 4 = 0 : On the line


Let consider two points (-1,-1). When we put this value on the equation of line we got 0. So we can say that this point is on the hyperplane of the line. 


Case 2: 
Similarly, 

x1 + 3x2 + 4 > 0 : Positive half-space


Consider two points (1,-1). When we put this value on the equation of line we got 2 which is greater than 0. So we can say that this point is on the positive half space. 
Case 3: 

x1 + 3x2 + 4 < 0 : Negative half-space


Consider two points (1,-2). When we put this value on the equation of line we got -1 which is less than 0. So we can say that this point is on the negative half-space. 
 


Last Updated : 03 Jul, 2020
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