I am currently reading Plane Alegbraic Curves by Andreas Gathmann. This book defines the projective $n$-space (which is denoted by $\mathbb{P}^n$) over a field $K$ as the set of all $1$-dimensional vector spaces of $K^{n+1}$ (where $K^{n+1}$ is regraded as a vector space over $K$). The book then characterizes $\mathbb{P}^n$ as follows: Define a relation $\rho$ on $K^{n+1}\setminus \lbrace 0 \rbrace$ as follows \begin{align} (x_0,\ldots,x_n) \rho (y_0,\ldots,y_n) \iff \exists \lambda \in K^* : (x_0,\ldots,x_n) = \lambda (y_0,\ldots,y_n). \end{align} Then the books says that $\mathbb{P}^n = (K^{n+1} \setminus \lbrace 0 \rbrace )/ \rho$. I wonder how this is possible?. Clearly, every equivalence class $[(x_0,\ldots,x_n)]_{\rho}$ fails even to be a vector subspace over $K$ since $0\not\in [(x_0,\ldots,x_n)]_{\rho}$ (The zero of $K^{n+1}$). Am I right?. If I`m mistaken, can anyone please help me understanding my mistake?
Thanks in advance.