A certain embedding of $\Bbb R^2_+$ onto its image in $\Bbb R^9$.

Question

Let $\Bbb R^{3^2} = M_{3\times 3}(\mathbb{R})$ and $φ(x,y)=(1,0,0,0,y,0,0,xy,y^2)$. Show that $φ:\mathbb{R^2_{+}}\longrightarrow{\mathbb{R^9}}$ is a embedding ($φ$ is an immersion and a homeomorphism onto $φ(\mathbb{R^2_{+}})\subset{\mathbb{R^9}}$)

Answer 1

Clearly $\varphi$ is smooth, and it is injective due to components $5$ and $8$ (using that $y>0$). Moreover the derivative $$D\varphi(x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ y & x \\ 0 & 2y \end{pmatrix}$$is injective due to rows $5$ and $8$ (again using that $y>0$). It is a homeomorphism onto its image because the inverse $\varphi^{-1}\colon \varphi(\Bbb R^2_+) \to \Bbb R^2_+$ given by $\varphi^{-1} = \psi|_{\varphi(\Bbb R^2_+)}$ is continuous, where $\psi\colon \Bbb R^9 \setminus \{u_5 = 0\} \to \Bbb R^2$ given by $$\psi(u_1,\ldots, u_9) = (u_8/u_5, u_5) $$is continuous as well.