In mathematics, projectivization is a procedure which associates with a non-zero

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

''V'' a projective space

(V)

, whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multiplication defines a subset of

(V)

formed by the lines contained in ''S'' and is called the projectivization of ''S''.

Properties

* Projectivization is a special case of the

factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...

by a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...

: the projective space

(V)

is the quotient of the open set ''V''\ of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

(V)

in the sense of algebraic geometry is one less than the dimension of the vector space ''V''. * Projectivization is

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

ial with respect to

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

linear maps: if ::

f: V\to W

: is a linear map with trivial

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...

then ''f'' defines an algebraic map of the corresponding projective spaces, ::

\mathbb(f): \mathbb(V)\to \mathbb(W).

: In particular, the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...

GL(''V'') acts on the projective space

(V)

by automorphisms.

Projective completion

A related procedure embeds a vector space ''V'' over a field ''K'' into the projective space

(V\oplus K)

of the same dimension. To every vector ''v'' of ''V'', it associates the line spanned by the vector of .

Generalization

In algebraic geometry, there is a procedure that associates a

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

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

{\mathbb P}(V).

This

Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not fun ...

gives rise to a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from the category of graded commutative rings and surjective graded maps to the category of projective schemes. Projective geometry Linear algebra