Exercise 3.65

Solution to Exercise 3.65.
It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said. "Yes, it is true, but you won't be able to understand the proof."
—Ronald Graham, Scientific American (1993)

Exercise 3.65. (a) Verify part (b) of Proposition 3.13.

Proof. Let A be the matrix of an affine direct isometry. Then X is an invariant point of the affine direct isometry if and only if

if and only of

if and only if the system of equations

has a unique solution x₁ and x₂ if and only if

if and only if

Hence the affine direct isometry has no invariant points if and only if i.e.

which is the matrix of a translation.//

(b) Verify part (b) of Proposition 3.14.

Proof. Let A be the matrix of an affine direct isometry. First, assume the invariant point of the affine direct isometry is the origin. Then

if and only of

if and only if

which is the matrix of an affine rotation about the origin.

Next, assume the invariant point C is not at the origin. Consider the transformation defined by . Since the set of direct isometries is a group, it is a direct isometry. Further, that is, the origin O is an invariant point. By the previous paragraph, there is a rotation such that Thus, is the matrix of an affine rotation about C.//

Proof. Let be an affine rotation of the Euclidean plane with center C. Consider We have

Further, is a direct isometry since the set of direct isometries is a group. Since direct isometries preserve distance and angle measure, for every point X in the Euclidean plane. Therefore, //