Linear Algebra/Topic: Orthonormal Matrices/Solutions
- Problem 1
Decide if each of these is an orthonormal matrix.
- No, the columns do not have length one.
- Problem 2
Write down the formula for each of these distance-preserving maps.
- the map that rotates radians, and then translates by
- the map that reflects about the line
- the map that reflects about and translates over and up
Some of these are nonlinear, because they involve a nontrivial translation.
- The line makes an angle of with the -axis. Thus and .
- Problem 3
- 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.
- Prove that congruence is an equivalence relation between plane figures.
- 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.
- 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).
The first two components of each are and .
- Problem 5
- 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 distance-preserving maps.
- 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 .
- 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.
- 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.