Is the union of dual cone and polar cone of a convex cone is a vector space?

Question

There is an exercise in the book Matrix Algebra that ask to show if $C$ is a convex cone, then the union $C^* \cup C^0$ is a vector space. Where $C^*$ is dual cone and $C^0$ is polar cone of $C$.

I think I find a counterexample that is depicted in the figure. The blue and orange vectors (and their combination) form a convex cone. I think the green and the purple regions are its dual and polar cone, respectively. The union of $C^*$ and $C^0$, cover all the space except white regions. Since there are vectors in the union that their linear combination locates in the white region (e.g. the black vector), the set is not closed under vector addition. Therefore it is not a subspace.

Where is my fault?

Thanks.

Answer 1

The original exercise was wrong, and the author has corrected this in an errata

"is a vector space" should be "is a cone but not necessarily a convex cone".

Your counterexample correctly shows that the union is not always a subspace (or even a convex cone), so it answers the corrected exercise!