Affine transformations of the plane preserve ratios of areas; that is, if $Area(F) = k \cdot Area(G)$, then $Area(T(F)) = k \cdot Area(T(G))$. They do not, however, preserve ratios of length, unless the two segments in question are colinear, in which case ratios of length are preserved.
Can this be generalized? If we think about n-dimensional hypervolume, with 1-hypervolume being length, 2-hypervolume area, etc., can we say:
For any figures in the same n-dimensional affine subspace, affine transformations preserve the ratio of n-hypervolume. That is, two the ratio of length of colinear line segments, the ratio of area of coplanar figures, the ratio of volume of solids in the same 3-dimensional flat, etc.
Best Answer
Yes, and this property follows immediately from applying the multivariable Chain Rule to a general affine transformation. Under the usual identifications, an affine transformation $F : {\bf x} \mapsto A {\bf x} + b$ has derivative $T_{\bf v} F = A$, so for any measurable region $U \subset \Bbb V$ the Chain Rule gives $$\operatorname{vol} F(U) = \int_{F(U)} dV = \int_{U} \det (TF) \,dV = \int_U \det A \,dV = \det A \int_U dV = \det A \cdot \operatorname{vol}(U) ,$$ where $dV$ denotes the volume form on $\Bbb V$. (Notice that this argument uses that $\det TF = A$ is constant, a property that characterizes affine maps $\Bbb V \to \Bbb V$.) So for all $U$ the ratio of signed volumes is $\det A$ and the ratio of unsigned volumes is $\vert\det A\vert$.