Exercise 4 — Linear Algebra

In practice the matrix for the distance-preserving linear transformation and the translation are often combined into one. Check that these two computations yield the same first two components.

\begin{pmatrix}a&c\\ b&d\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}e\\ f\end{pmatrix}\qquad\begin{pmatrix}a&c&e\\ b&d&f\\ 0&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ 1\end{pmatrix}

(These are homogeneous coordinates; see the Topic on Projective Geometry).

Hefferon, Linear Algebra, Maps Between Spaces, Orthonormal Matrices, Exercise 4

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