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A collineation is a one-to-one map from one projective space to another, or from a projective plane onto itself, such that the images of collinear points are themselves collinear. All automorphisms induce a collineation.
## Definition

Let V be a vector space (of dimension at least three) over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W).
Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:## Fundamental theorem of projective geometry

Briefly, every collineation is the product of a homography and an automorphic collineation. In particular, the collineations of PG(2, R) are exactly the homographies.## See also

- α is a bijection.
- A ⊆ D ↔ A
^{α}⊆ B^{α}for all A, B in D(V).

When V has dimension one, a collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that :

- 0 is mapped onto the trivial subspace of W.
- V is mapped onto W.
- There is a nonsingular semilinear map β from V to W such that, for all v in V,

- $(langle\; vrangle)^\{alpha\}=langle\; v^\{beta\}rangle$

The reason for the seemingly completely different definition when V has geometric dimension one will become clearer further on in this article.

When V = W the collineations are also called automorphisms.

Suppose φ is a semilinear nonsingular map from V to W, with the dimension of V at least three. Define α : D(V) → D(W) by saying that Z^{α} = { φ(z) | z ∈ Z } for all Z in D(V). As φ is semilinear, one easily checks that this map is properly defined, and further more, as φ is not singular, it is bijective. It is obvious now that α is a collineation. We say α is induced by φ.

The fundamental theorem of projective geometry states the converse:

Suppose V is a vector space over a field K with dimension at least three, W is a vector space over a field L, and α is a collineation from PG(V) to PG(W). This implies K and L are isomorphic fields, V and W have the same dimension, and there is a semilinear map φ such that φ induces α.

The fundamental theorem explains the different definition for projective lines. Otherwise, every bijection between the points would be a collineation, and then there would be no nice algebraic relationship.

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Last updated on Monday March 31, 2008 at 17:30:04 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday March 31, 2008 at 17:30:04 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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