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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*.^{[1]}^{[2]}

- 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

- is a linear map with trivial kernel then
*f*defines an algebraic map of the corresponding projective spaces,

- In particular, the general linear group GL(
*V*) acts on the projective space**P**(*V*) by automorphisms.

A related procedure embeds a vector space *V* over a field *K* into the projective space **P**(*V* ⊕ *K*) of the same dimension. To every vector *v* of *V*, it associates the line spanned by the vector (*v*, 1) of *V* ⊕ *K*.

In algebraic geometry, there is a procedure that associates a projective variety Proj *S* with a graded commutative algebra *S* (under some technical restrictions on *S*). If *S* is the algebra of polynomials on a vector space *V* then Proj *S* is **P**(*V*). This Proj construction gives rise to a contravariant functor from the category of graded commutative rings and surjective graded maps to the category of projective schemes.