Let $V$ be a $K$-vector space, $\mathbb{P}(V)$ the projective space derived from $V$, $\pi:V-\{0\}\rightarrow\mathbb{P}(V)$ the canonical surjection and $(e_i)_{0\leq i\leq n}$ a basis of $V$.
Suppose $W$ is a linear subspace of $V$ of dimension $r+1$. Then it is asserted that $\pi(W-0)\cong\mathbb{P}(W)$ can be defined by
a system of $n-r$ homogeneous linear equations $$\sum_{i=0}^n\xi_i\alpha_{ij}=0,\,(1\leq j\leq n-r)$$ between homogeneous coordinates $(\xi_i)_{0\leq i\leq n}$ of a point of $\mathbb{P}(V)$ w.r.t. the basis $(e_i)_{0\leq i\leq n}$, the left hand sides being linearly independent forms on $V$.
I can't understand this at all; what is being asserted here?–Let me state what I know. Let $(x^*_j)_{1\leq j\leq n-r}$ be a basis of the orthogonal $W'$ of $W$ in $V^*$. Then $$W=\{x\in V\ |\langle x,x^*_j\rangle=0\text{ for all }1\leq j\leq n\}.$$ That is to say, $W$ is defined by the system of equations $$\sum_{i=0}^n\alpha_i\langle e_i,x^*_j\rangle=0,\,(1\leq j\leq n-r).$$ Apart from this, what else is the quoted result stating? What does it mean to say that $\pi(W-0)$ is defined by such a system of linear equations?