# Projective transformation

A one-to-one mapping $F$ of a projective space $\Pi _ {n}$ onto itself preserving the order relation in the partially ordered (by inclusion) set of all subspaces of $\Pi _ {n}$, that is, a mapping of $\Pi _ {n}$ onto itself such that:

1) if $S _ {p} \subset S _ {q}$, then $F ( S _ {p} ) \subset F ( S _ {q} )$;

2) for every $\widetilde{S} _ {p}$ there is an $S _ {p}$ such that $F ( S _ {p} ) = \widetilde{S} _ {p}$;

3) $S _ {p} = S _ {q}$ if and only if $F ( S _ {p} ) = F ( S _ {q} )$.

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, a perspective and a homology.

Let the space $\Pi _ {n}$ be interpreted as the collection of subspaces $P _ {n} ( K )$ of the left vector space $A _ {n+} 1 ( K )$ over a skew-field $K$. A semi-linear transformation of $A _ {n+} 1$ into itself is a pair $( \overline{F}\; , \phi )$ consisting of an automorphism $\overline{F}\;$ of the additive group $A _ {n+} 1$ and an automorphism $\phi$ of the skew-field $K$ such that for any $a \in A _ {n+} 1$ and $k \in K$ the equality $\overline{F}\; ( ka ) = \phi ( k ) \overline{F}\; ( a )$ holds. In particular, a semi-linear transformation $( \overline{F}\; , \phi )$ is linear if $\phi ( k) \equiv k$. A semi-linear transformation $( \overline{F}\; , \phi )$ induces a projective transformation $F$. The converse assertion is the first fundamental theorem of projective geometry: If $n \geq 2$, then every projective transformation $F$ is induced by some semi-linear transformation $( \overline{F}\; , \phi )$ of the space $A _ {n+} 1 ( K )$.

#### References

 [1] R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) MR0052795 Zbl 0049.38103 [2] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502

A projective transformation can also be defined as a bijection of the points of $\Pi _ {n}$ preserving collinearity in both directions.