Projectivization

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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]

• Projectivization is a special case of the
  dimension of P(V) in the sense of algebraic geometry
  is one less than the dimension of the vector space V.
• Projectivization is
  injective
  linear maps: if

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

${\displaystyle \mathbf {P} (f):\mathbf {P} (V)\to \mathbf {P} (W).}$

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