Orthonormal Matrices

Decide if each of these is an orthonormal matrix.

• $\begin{pmatrix}1/\sqrt{2}&-1/\sqrt{2}\\ -1/\sqrt{2}&-1/\sqrt{2}\end{pmatrix}$

• $\begin{pmatrix}1/\sqrt{3}&-1/\sqrt{3}\\ -1/\sqrt{3}&-1/\sqrt{3}\end{pmatrix}$

• $\begin{pmatrix}1/\sqrt{3}&-\sqrt{2}/\sqrt{3}\\ -\sqrt{2}/\sqrt{3}&-1/\sqrt{3}\end{pmatrix}$

Write down the formula for each of these distance-preserving maps.

• the map that rotates $\pi/6$ radians, and then translates by $\vec{e}_{2}$

• the map that reflects about the line $y=2x$

• the map that reflects about $y=-2x$ and translates over $1$ and up $1$

• The proof that a map that is distance-preserving and sends the zero vector to itself incidentally shows that such a map is one-to-one and onto (the point in the domain determined by $d_{0}$, $d_{1}$, and $d_{2}$ corresponds to the point in the codomain determined by those three). Therefore any distance-preserving map has an inverse. Show that the inverse is also distance-preserving.

• Prove that congruence is an…

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.

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

• Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so.

• Give two more properties that are of interest in Euclidean geometry from your experience in studying that subject that are also invariant under distance-preserving maps.

• Give a property that is not of interest in Euclidean geometry and is not invariant under d…