Projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S. == Properties == Projectivization is a special case of the factorization by a group action: the projective space P(V) is the quotient of the open set V \ {0} of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of P(V) in the sense of algebraic geometry is one less than the dimension of the vector space V. Projectivization is functorial with respect to injective linear maps: if f : V → W {\displaystyle f:V\to W} is a linear map with trivial kernel then f defines an algebraic map of the corresponding projective spaces, P ( f ) : P ( V ) → P ( W ) .