Definition. A map from N-dimensional Cartesian space to itself is called a transformation if it is bijective, i.e., both ``one-to-one'' and ``onto''. (This definition does not require ``continuity'' although definitions in other contexts might.)
Definition. A map from N-dimensional Cartesian space to M-dimensional Cartesian space is an affine map if it has the form
f(x) = A x + b ,
where A is a matrix of the appropriate size.Proposition. An affine map preserves barycentric combinations in the following sense:
If x = SUM_{j}[ u_{j} x_{j} ] with SUM_{j}[ u_{j} ] = 1 , then f(x) = SUM_{j}[ u_{j} f(x_{j}) ] .
Definition. An affine transformation of N-dimensional Cartesian space is a transformation that is also an affine map.
Theorem. An affine map from N-dimensional Cartesian space to itself is an affine transformation if and only if the associated N \times N matrix is an invertible matrix. In this case the inverse map is an affine transformation whose associated matrix is the inverse of the matrix associated with the original affine transformation.
Let A, B, and C be the points in the Cartesian plane that are given by
A = (0, -1) B = (3, 4) and C = (-1, 1) ,
and let T be the triangle with vertices A, B, and C.
Find an affine transformation f of the plane for which f(0, 0) = A, f(1, 0) = B, and f(0, 1) = C.
Is more than one solution of the preceding exercise possible?
Let f and g be the affine maps from the Cartesian plane R^{2} to itself that are defined by
f(x) = R x + r and g(x) = S x + s .
Compute g \circ f and f \circ g, where `\circ ' denotes composition of maps.Could the plane be replaced by 3-dimensional space in the preceding exercise? What about N-dimensional space?
Prove that an affine map is a linear map in the sense of second year undergraduate ``linear algebra'' if and only if it carries the origin of its domain to the origin of its target.