Terminology Revision. Any weight 1 linear combination of given points may be called a barycentric combination of those points, regardless of whether the coefficients are non-negative.
Definition. A sequence of r + 1 points p_{0}, p_{1}, ..., p_{r} is called barycentrically independent if none of them is a barycentric combination of the others.
Examples.
Any two distinct points P, Q are barycentrically independent. If P <> Q, the set of barycentric combinations of P and Q is the line through P and Q.
Three points A, B, C are barycentrically independent if and only if none lies on the line determined by the other two. Thus, the vertices of a triangle are barycentrically independent.
In R^{3} the four vertices of a tetrahedron are barycentrically independent.
Proposition. A sequence of r + 1 points p_{0}, ..., p_{r} is barycentrically independent if and only for given a_{0}, ..., a_{r} and given b_{0}, ..., b_{r} with a_{0} + ... + a_{r} = 1 and b_{0} + ... + b_{r} = 1 the following statement is true:
| a_{0} p_{0} + ... + a_{r} p_{r} = b_{0} p_{0} + ... + b_{r} p_{r} if and only if a_{0} = b_{0}, ... , a_{r} = b_{r} . |
Proof. Obtain this from corresponding facts about linear independence.
Theorem. If p_{0}, p_{1}, ..., p_{n} are barycentrically independent points of n-dimensional Euclidean space R^{n}, and q_{0}, q_{1}, ..., q_{n} are any points of R^{m}, then there is one and only one affine map f from R^{n} to R^{m} for which f(p_{0}) = q_{0}, f(p_{1}) = q_{1}, ..., f(p_{n}) = q_{n}.
Proof. Use the fact that there is a unique linear map taking prescribed values at the members of a basis of R^{n}.
Theorem If a map f from R^{n} to R^{m} preserves barycentric combinations, then it must be an affine map.
Proof. Use two facts: (1) an affine map that carries 0 to 0 must be linear, and (2) a linear map is always given by a matrix.
Let A, B, C, and D be four points in the plane R^{2}. Show that the polygonal path (sequence of line segments) from A to B, from there to C, then to D, and back to A is a parallelogram if and only if A - B + C - D = 0.
Show that an affine transformation of the plane carries a parallelogram to a parallelogram.
Show that there is one and only one affine transformation of the plane carrying a given parallelogram to another given parallelogram in a given vertex-matching way.
Show that any affine transformation of the plane carries the point where the diagonals of a given parallelogram meet to the point where the diagonals of the image parallelogram meet.
Explain why an affine transformation of the 3-dimensional space R^{3} must always carry a tetrahedron to a tetrahedron.