The Dines theorem and some other properties of quadratic mappings

Real homogeneous quadratic mappings from $\mathbb{R}^n$ to $\mathbb{R}^2$ are examined. It is known that the image of such a mapping is always convex. A proof of the convexity of the image based on the quadratic extremum principle is given. The following fact is noted: If the quadratic mapping $Q$ is surjective and $n>2+\mathrm{dim\,ker\,}Q$, then there exists a regular zero of $Q$. A certain criterion of the linear dependence of quadratic forms is also stated.