**Vector Space Construction**

The line at infinity of the extended real plane appears to have a different nature than the other lines of that projective plane. This, however, is not true. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, the "points" of the real projective plane are the lines through the origin in 3-dimensional Euclidean space, and a "line" in the projective plane arises from a plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.

Let *K* be any division ring (skewfield). Let *K*3 denote the set of all triples *x* = (*x*_{0}, *x*_{1}, *x*_{2}) of elements of *K* (a Cartesian product viewed as a Vector space). For any nonzero *x* in *K*3, the *line* in *K*3 through the origin and *x* is the subset

of *K*3. Similarly, let *x* and *y* be linearly independent elements of *K*3, meaning that if *k x + l y = 0* then *k = l = 0*. The *plane* through the origin, *x*, and *y* in *K*3 is the subset

of *K*3. This plane contains various lines through the origin which are obtained by fixing either k or l.

The **projective plane** over *K*, denoted PG(2,*K*) or *K***P**2, has a point set consisting of all the lines in *K*3 through the origin (each is a vector subspace of dimension 1). A subset *L* of PG(2,*K*) is a *line* in PG(2,*K*) if there exists a plane in *K*3 whose set of lines is exactly *L* (a vector subspace of dimension 2).

Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.

An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set *K*3 - {(0, 0, 0)} modulo the equivalence relation

- x ~ k x, for all k in .

Lines in the projective plane are defined exactly as above.

The coordinates (*x*_{0}, *x*_{1}, *x*_{2}) of a point in PG(2,*K*) are called **homogeneous coordinates**. Each triple (*x*_{0}, *x*_{1}, *x*_{2}) represents a well-defined point in PG(2,*K*), except for the triple (0, 0, 0), which represents no point. Each point in PG(2,*K*), however, is represented by many triples.

If *K* is a topological space, then *K***P**2, inherits a topology via the product, subspace, and quotient topologies.

Read more about this topic: Projective Plane

### Other articles related to "vector space construction, vector space":

**Vector Space Construction**- Desargues' Theorem and Desarguesian Planes

... plane if and only if the plane can be constructed from a 3 dimensional

**vector space**over a skewfield as above ...

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