What could be an intutive understanding of a hyperplane? This "Hyperplane" word gradually becoming more important to understand as I delve deeper into machine learning applications.
To explain Hyperplane, the wiki article majorly speaks about one dimension less than the dimension occupied by space. https://en.wikipedia.org/wiki/Hyperplane
However I could not build a intuitive understanding from this.
The earlier similar attempts for related questions in SO also could not help much in this aspect.
https://stackoverflow.com/questions/24432871/what-is-the-difference-between-hyperplane-and-plane-and-why-is-hyperplane-repr
and
hyperplane in svm
It would be of much help if some sort of visual or analytical  explanation can be provided from the experts.
Thanks for your time.
 A: Maybe here's a simple example to demystify high-dimensional space.
Imagining a blood test that gives you the result of one parameter, say haemoglobin concentration. You can plot the results from many patients as points on a line and see if they form clusters.
Now consider a blood test that gives two parameters, say haemoglobin and hematocrit. Now you can plot patients on a 2D plot and look for clusters (you see where this is going). For a test that gives you three parameters, (haemoglobin,  hematocrit, RBC) you can use a 3D plot.
A blood test that gives you 20 parameters (haemoglobin,  hematocrit, RBC, WBC, etc) cannot be visualised, still such test is no more esoteric than the previous ones and linear algebra handles it just fine.
A: You may think Hyperplane is a linear "decision boundary" on high dimensional space.
We can start with 1D and add it up to build up the intuition:

*

*When D=1, an example of hyperplane can be x=0. So, the "decision boundary" is a point. And we can use this decision point, to classify any real number into 2 classes.


*When D=2, an example of hyperplane can be x+y=0. So, the "decision boundary" is a line. And we can use this decision line, to classify any point in 2D into 2 classes.


*When D=3, an example of hyperplane can be x+y+z=0. So the "decision boundary" is a plane. And we can use this decision plane, to classify any point in 3D into 2 classes.
If we plot up to D=3, we can see, point, line, and plane.

When D>3, it is not easy to plot. But we still can look at the equation. For example, when D=5, a hyperplane can be x+y+z+p+q=0. Just an equation to describe a linear relationship among more variables.
