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Linear Maps
A map which is
additive:
homogenous:
Representable by a transformation matrix :

Linear map preserve the origin.

Affine Maps
A map which is
composed of a linear map and a translation:

Affine transformations can be represented as a matrix-vector product by using extended coordinates:

Affine maps preserve collinearity of points and distance ratios.

Projective Maps
Represented by a matrix-vector product using homogeneous coordinates.
Projective maps preserve cross ratios: