How do you define a space of a cone in linear algebra?
Let A be a $m \times r$ matrix, the cone generated by its columns is:
$$ C_{A} = \{ Ax \mid x \geq 0\} \subseteq \mathbb{R}^m$$
My question is, why is the word cone used to define this? I tried an example in $\mathbb{R}^2$ to get an intuition of this and it made sense to me, because if we only have two independent vectors in $\mathbb{R}^2$ then we basically cover a triangle (which I guess is the equivalent of a cone in 2D?).
Anyway, in 3D it made less sense to me because I expected a circular cone to be covered, but it seemed more like a 3D pyramid. Is that correct? Or maybe what I need is a rigorous definition of a cone and see if $C_{A}$ satisfies that definition. Does someone have an idea of why the word cone was used and how to justify it? Either rigorously or intuitively?
Edit:
after some googling it seems thats the definition of cone in linear algebra is:
In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars
Kind of a unsatisfying definition because it didn't really look like a cone in $\mathbb{R}^3$, more like a triangular based pyramid (if we have the rank of A to be 3). Am I wrong? Or why is that called a cone?
$\endgroup$ 10 Reset to default
