Linear Algebra/Topic: Orthonormal Matrices/Solutions

SolutionsEdit

Problem 1

Decide if each of these is an orthonormal matrix.

  1.  
  2.  
  3.  
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.  
  2. The line   makes an angle of   with the  -axis. Thus   and  .
     
  3.  
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.