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
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If we expand this out for n variables we will get something like this
X1n1 + X2n2 + X3n3 + ……….. + Xnnn + b = 0
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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 with2. Subspace : Hyper-planes, in general, are not sub-spaces. However, if we have hyper-planes of the form,Though it's a 2D geometry the value of X will be So according to the equation of hyperplane it can be solved as So as you can see from the solution the hyperplane is the equation of a line.
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 asx1 + 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.