Why is a projective subspace itself a projective variety?

Question

Let K^(n+1) be a vector space and W be its subspace.

Then projective subspace P(W) is said to be a projective variety of P(K^(n+1)). Could anyone tell me why it is so?

Answer 1

A line in $W$ is a line in $K^{n+1}$ by the inclusion $W\to K^{n+1}$; this realises $P(W)$ as a subset of $P(K^{n+1})$.

To see that this subset is algebraic, interpret a linear equation defining $W$ in $K^{n+1}$ (which is homogeneous of degree $1$) as an equation in the homogeneous coordinates on $P(K^{n+1})$.