The image of an affine transformation

Compute affine images of points and sets, using rank and determinant to explain dimension, area scale, and orientation.

Images of points and sets

A point \(\mathbf p\) has image \(T(\mathbf p)=A\mathbf p+\mathbf b\). For a set \(S\), the image \(T(S)\) contains exactly those outputs obtained as \(\mathbf p\) ranges over \(S\).

Because affine combinations are preserved, a polygon or simplex can be mapped by transforming its vertices and retaining the same convex-combination parameters. Degeneracy may cause distinct vertices or edges to coincide.

Rank controls output dimension

Across the whole domain, \(A\mathbf x\) ranges over \(\operatorname{Col}(A)\), so \(T(\mathbb R^n)=\mathbf b+\operatorname{Col}(A)\). This affine range has dimension \(\operatorname{rank}(A)\).

For \(A\in\mathbb R^{2\times2}\), rank two can fill the plane, rank one confines outputs to a line, and rank zero maps every input to the single point \(\mathbf b\).

Worked example: a square collapses

Let \(A=\begin{bmatrix}1&2\\2&4\end{bmatrix}\), \(\mathbf b=[1,-1]^T\), and let \(S=[0,1]^2\). The four vertex images are \((1,-1)\), \((2,1)\), \((3,3)\), and \((4,5)\).

All satisfy \(y=2x-3\), and \(\operatorname{rank}(A)=1\). The filled square therefore maps to the line segment from \((1,-1)\) to \((4,5)\), not to a two-dimensional parallelogram.

Preserved structure and limits

Affine maps send lines to lines or to points, preserve ratios along a line, and preserve parallel directions when those directions are not collapsed. They do not generally preserve Euclidean lengths or angles.

Mapping only the boundary can be misleading under rank loss unless the full convex set is understood. The formula \(T((1-t)p+tq)=(1-t)T(p)+tT(q)\) tracks every point on each segment.

Area scale and orientation in the plane