Why is any subspace a convex cone? I am reading Convex Optimization written by Stephen Boyd. In page 27 of chapter 2, there is an example said 'Any subspace is affine, and a convex cone(hence convex).'
Can anybody explain to me why this is true?
 A: First, what basically distinguishes the definitions of convex, affine and cone, is the domain of the coefficients and the constraints that relate them.
Let us starts by the first part:
any subspace is affine, which means, if we have:
$x_1, x_2 \in V$, where $V$ is a subspace; therefore any linear combination of these two vectors must lie in $V$. That is, if we have two coefficients $\theta_1, \theta_2 \in \mathcal{R}$, then, $\theta_1x_1 + \theta_2x_2 \in V$.
The definition of affine sets tells us if $x_1,x_2$ are in an affine set, their linear combination must also lie in the same set, with the condition the coefficients must sum to 1, that is $\theta_1 + \theta_2 = 1$. Now, assume we have chosen $\theta_2 = 1- \theta_1$, therefore the combination $\theta_1x_1 + (1 - \theta_1)x_2 \in V$
Therefore any subspace is affine, since we have the freedom to choose the coefficients to sum to 1.
Now why a subspace is a convex cone.
Notice that, if we choose the coeficientes $\theta_1, \theta_2 \in \mathcal{R}_+$, we actually define a cone, and if the coefficients sum to 1, it is convex, therefore it is a convex cone.
A: because a linear subspace contains all multiples of its elements as well as all linear combinations (in particular convex ones). Hence it is a convex cone.
