Solution to Exercise 3.41.
A mathematics teacher who talks at length affects both ends of students. One end is made to feel numb and the other end dumb.
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Exercise 3.41. Prove Proposition 3.9.
Proposition 3.9. The set of affine direct isometries of the Euclidean plane is a group.

 

Proof. By Proposition 3.8, the set of affine direct isometries is closed under matrix multiplication. Hence, we only need to show the inverse of a direct isometry is a direct isometry. Let  be the matrix of an affine direct isometry of the Euclidean plane.

 

Since a₂₁ = a₁₂, A^–1 is the matrix of an affine direct isometry of the Euclidean plane. Hence, the inverse of an affine direct isometry of the Euclidean plane is an affine direct isometry of the Euclidean plane. Therefore, the set of affine direct isometries of the Euclidean plane is a group.//