Show isometry and find geometric meaning

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In summary, the matrix ##F## in question is a rotation about the line ##t(1,1,1)## followed by reflection in the origin.
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TL;DR Summary
With respect to an orthonormal basis in 3-space, show that the matrix of ##F## given below is an isometry and find its geometric meaning.
The matrix ##A## in question is

##\dfrac{1}{3}
\left(\begin{array}{rrr}
-2 & 1 & -2 \\
-2 & -2 & 1 \\
1 & -2 & -2
\end{array}\right)##
One can easily verify that ##AA^t=I##, hence an isometry. To find its geometric meaning, one can proceed to find ##U=\text{ker} \ (F-I)=\text{ker} \ (A-I)##, i.e. ## (A-I)\textbf{x}=\textbf{0}##. This yields ##\textbf{x}=\textbf{0}##. Put ##G=-F## and find ##U=\text{ker} \ (G-I)=\text{ker} \ (A+I)## instead, which yields ##\textbf{x}=t(1,1,1)##. Thus ##G## is a rotation about the line ##t(1,1,1)##. To determine the orientation of the rotation, one can define ##\textbf{w}=(1,1,1)##, ##\textbf{u}=(1,-1,0)## (any vector perpendicular to ##\textbf{w}##) and ##\textbf{v}=A\textbf{u}=(-1,0,1)##. Evaluating the determinant with the columns given by ##\textbf{u},\textbf{v},\textbf{w}## shows that they are negatively oriented and thus the rotation is clockwise seen from the point ##(1,1,1)## down on the origin. The angle of rotation is deduced from the dot product of ##\textbf{u}## and ##\textbf{v}##, which is ##\frac{2\pi}{3}##.

This is the geometric meaning of ##G##. What is that of ##F## and its matrix? The answer given is a rotation ##\frac{\pi}{3}## anti-clockwise from the point ##(1,1,1)## followed by reflection in the origin. Since ##G=-F##, isn't ##F## the rotation of ##G## followed by reflection in the origin? That is, a clockwise rotation ##\frac{2\pi}{3}## from the point ##(1,1,1)## looking down on the origin followed by reflection. The answer seems to reverse this.

If ##G = -F##, let ##v \in \mathbb{R}^3##. Geometrically, ##Fv = -Gv## and thus ##v## transforms first under ##G## and the result ##Gv## gets an opposite sign, meaning that we reflect the vector ##Gv## around the origin.

But the order shouldn't matter actually: since the maps involved are linear, we also have ##Fv = G(-v)## and then ##F## first reflects the vector ##v## around the origin and then performs a rotation.

Okay, so the answers are equivalent?

Could it be that the answer tries to describe the transformation ##G## in terms of ##F##? Then it would make sense. However, that wouldn’t be the geometric meaning of ##F## and its matrix.

1. What is an isometry?

An isometry is a type of transformation in geometry that preserves the size and shape of a figure. It can include translations, rotations, and reflections.

2. How do you show isometry?

To show isometry, you must demonstrate that the transformed figure is congruent to the original figure. This can be done by comparing corresponding sides and angles of the two figures.

3. What is the geometric meaning of isometry?

The geometric meaning of isometry is that the transformed figure is an exact copy of the original figure, just in a different position or orientation.

4. How does isometry differ from similarity?

While isometry preserves both size and shape, similarity only preserves shape. This means that the corresponding angles of similar figures are equal, but the corresponding sides may be different lengths.

5. Why is isometry important in geometry?

Isometry is important in geometry because it allows us to analyze and compare figures without changing their fundamental properties. It also helps us understand the relationship between different geometric shapes and their transformations.

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