An affine transformation of R^{n} is called orientation-preserving if the determinant of its associated matrix is positive and is called orientation-reversing if the determinant of its associated matrix is negative.
An ordered sequence B_{0}, B_{1}, ..., B_{n} of n + 1 barycentrically independent points in n-dimensional Cartesian space R^{n} determines an affine coordinate system in R^{n} as follows: if P is a point and if
i.e., if x_{0}, x_{1}, ..., x_{n} are the barycentric coordinates of P with respect to B_{0}, B_{1}, ..., B_{n}, then the affine coordinates of P with respect to B_{0}, B_{1}, ..., B_{n} are x_{1}, x_{2}, ..., x_{n}. This has the effect of declaring B_{0} to be the origin and B_{1}, ..., B_{n} to be the unit points on the positive coordinate axes. One says that x = (x_{1}, x_{2}, ..., x_{n}) is the coordinate vector of P in this affine coordinate system.
Theorem. If B_{0}, B_{1}, ..., B_{n} and C_{0}, C_{1}, ..., C_{n} are two given ordered sequences of n + 1 barycentrically independent points in R^{n}, then there is an invertible n \times n matrix R and a vector r in R^{n} such that for every point P its affine coordinate vector x in the one affine coordinate system and its affine coordinate vector y in the other coordinates system are related by the formula y = R x + r. Abbreviated statement: An affine change of coordinates always has the form y = R x + r where R is an invertible matrix and r is a vector.
Show that the barycentric coordinates of a point in the plane relative to a fixed ordered triple of non-collinear points do not depend on the choice of an affine coordinate system for the plane.
Show that a 2 \times 2 orthogonal matrix must have one of the forms
| cos t | - sin t |
| sin t | cos t |
| cos t | sin t |
| sin t | - cos t |
Let A, B, C,and D be four points in the plane. Show that the quadrilateral ABCD (obtained by tracing the line segment AB, then the line segment BC, the line segment CD, and finally the line segment DA in that order) is a parallelogram if and only if A, B, and C are non-collinear and the ordered triple of barycentric coordinates of D with respect to the ordered triple A, B, C is the ordered triple 1, -1, 1.