I'm trying to wrap my head around the concept of hyperplanes, and this is what I've understood so far (correct me if I'm wrong):
- A hyperplane is a set of the form $\{x ~|~a^Tx=b \},~a\in I\!R^n,~x \in I\!R^n,~ b \in I\!R$
- b denotes the distance of the hyperplane from origin.
- x denotes points on the hyperplane that satisfy the equation
But I don't understand what a means. I know that it's a vector, but what does that signify geometrically with respect to the hyperplane? How to calculate it?
If someone can give numerical examples with geometric representations or point me to resources that provides such examples, that would be great.
A few more questions:
- What is the significance of the whole concept of hyperplanes, and why is it considered so important in some aspects of Linear Algebra and Machine Learning?
- What are some real world applications of this concept?
${\bf a}$ is the (green) vector from the origin that strikes the plane perpendicularly: