Projective subspace ($\mathbb{P}(W)\subseteq \mathbb{P}(V)$)

A projective subspace of $\mathbb{P}(V)$ is $\mathbb{P}(W)$ for a nonzero vector subspace $W\subseteq V$. Its projective dimension is $\dim W-1$. Thus one-dimensional vector subspaces represent projective points, while two-dimensional vector subspaces give projective lines.

For example, in ${\mathbb{P}}^3(K)$ the linear equation $x_3=0$ defines a projective plane. Adding the independent equation $x_2=0$ leaves a projective line. A consistent homogeneous linear system therefore describes a projective subspace through the nonzero vectors in its linear solution space.

If two corresponding vector subspaces have zero intersection, their projectivizations are disjoint; otherwise their intersection is $\mathbb{P}(W_1\cap W_2)$. The vector-space dimension formula explains why two projective lines in a plane always meet, whereas two lines in projective three-space can be skew. Nonlinear homogeneous equations instead define more general projective varieties.