Linear Algebra/Topic: Orthonormal Matrices/Solutions

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Solutions[edit]

Problem 1

Decide if each of these is an orthonormal matrix.

Answer
  1. Yes.
  2. No, the columns do not have length one.
  3. Yes.
Problem 2

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

  1. the map that rotates radians, and then translates by
  2. the map that reflects about the line
  3. the map that reflects about and translates over and up
Answer

Some of these are nonlinear, because they involve a nontrivial translation.

  1. The line makes an angle of with the -axis. Thus and .
Problem 3
  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 , , and 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.
  2. Prove that congruence is an equivalence relation between plane figures.
Answer
  1. Let be distance-preserving and consider . Any two points in the codomain can be written as and . Because is distance-preserving, the distance from to equals the distance from to . But this is exactly what is required for to be distance-preserving.
  2. Any plane figure is congruent to itself via the identity map , which is obviously distance-preserving. If is congruent to (via some ) then is congruent to via , which is distance-preserving by the prior item. Finally, if is congruent to (via some ) and is congruent to (via some ) then is congruent to via , which is easily checked to be distance-preserving.
Problem 4

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).

Answer

The first two components of each are and .

Problem 5
  1. Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so.
  2. 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.
  3. Give a property that is not of interest in Euclidean geometry and is not invariant under distance-preserving maps.
Answer
  1. The Pythagorean Theorem gives that three points are colinear if and only if (for some ordering of them into , , and ), . Of course, where is distance-preserving, this holds if and only if , which, again by Pythagoras, is true if and only if , , and are colinear. The argument for betweeness is similar (above, is between and ). If the figure is a triangle then it is the union of three line segments , , and . The prior two paragraphs together show that the property of being a line segment is invariant. So is the union of three line segments, and so is a triangle. A circle centered at and of radius is the set of all points such that . Applying the distance-preserving map gives that the image is the set of all subject to the condition that . Since , the set is also a circle, with center and radius .
  2. Here are two that are easy to verify: (i) the property of being a right triangle, and (ii) the property of two lines being parallel.
  3. One that was mentioned in the section is the "sense" of a figure. A triangle whose vertices read clockwise as , , may, under a distance-preserving map, be sent to a triangle read , , counterclockwise.