= Projective subspace
{title2=$\mathbb P(W)\subseteq\mathbb P(V)$}
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.
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