Projectivization

In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space ${\displaystyle {\mathbb {P}}(V)}$, whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of ${\displaystyle {\mathbb {P}}(V)}$ formed by the lines contained in S and is called the projectivization of S.

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

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