Currently, I am studying Boyd & Vandenberghe's Convex Optimization. From page 27:
2.2 Some Important Examples
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Any line is affine. If it passes through zero, it is a subspace, hence also a convex cone.
I am familiar with why a line is an affine set and that if it passes through the origin it is (by definition) a subspace. However, why does this make it a convex cone?
By definition, a set $C$ is a convex cone if for any $x_1, x_2 \in C$ and $\theta_1, \theta_2 \ge 0$,
$$\theta_1 x_1 + \theta_2 x_2 \in C$$
This makes sense and is easy to visualize. However, my understanding would be that a line passing through the origin would not satisfy the constraints put on $\theta$ because it can also go past the origin to the negative side (if that makes sense).
If $x_1, x_2$ are on the same line that passes through the origin, there exists $k_1, k_2 \in \mathbb{R}$ and a vector $v$ such that $$x_i = k_iv.$$
Let $\theta_1, \theta_2 \ge 0$.
Then $$\sum_{i=1}^2\theta_ix_i=\left(\sum_{i=1}^2 \theta_i k_i \right)v$$
which remains on the line.