View source for Projective transformation

A one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753801.png" /> of a [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753802.png" /> onto itself preserving the order relation in the partially ordered (by inclusion) set of all subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753803.png" />, that is, a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753804.png" /> onto itself such that:

1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753805.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753806.png" />;

2) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753807.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753808.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753809.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538010.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538011.png" />.

Under a projective transformation the sum and intersection of subspaces are preserved, points are mapped to points, and independence of points is preserved. The projective transformations constitute a group, called the projective group. Examples of projective transformations are: a [[Collineation|collineation]], a [[Perspective|perspective]] and a [[Homology|homology]].

Let the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538012.png" /> be interpreted as the collection of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538013.png" /> of the left vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538014.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538015.png" />. A semi-linear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538016.png" /> into itself is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538017.png" /> consisting of an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538018.png" /> of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538019.png" /> and an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538020.png" /> of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538021.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538023.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538024.png" /> holds. In particular, a semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538025.png" /> is linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538026.png" />. A semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538027.png" /> induces a projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538028.png" />. The converse assertion is the first fundamental theorem of projective geometry: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538029.png" />, then every projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538030.png" /> is induced by some semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538031.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538032.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Baer,   "Linear algebra and projective geometry" , Acad. Press  (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.V.D. Hodge,   D. Pedoe,   "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press  (1947)</TD></TR></table>



====Comments====
A projective transformation can also be defined as a bijection of the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538033.png" /> preserving collinearity in both directions.

Other names used for a projective transformation are: projectivity, collineation. See also [[Collineation|Collineation]] for terminology.