Linear Algebra/Topic: Orthonormal Matrices/Solutions

Solutions Problem 1 Decide if each of these is an orthonormal matrix. <li> <li> <li> Answer <li> Yes. <li> No, the columns do not have length one. <li> Yes. Problem 2 Write down the formula for each of these distance-preserving maps. <li> the map that rotates radians, and then translates by <li> the map that reflects about the line <li> the map that reflects about and translates over and up Answer Some of these are nonlinear, because they involve a nontrivial translation. <li> <li> The line makes an angle of with the -axis. Thus and . <li> Problem 3exer:IsometryFacts <li> 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. <li> Prove that congruence is an equivalence relation between plane figures. Answer <li> 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. <li> 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 4exer:HomoCrds 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. (homogeneous coordinatesThese are homogeneous coordinates; see the Topic on Projective Geometry). Answer The first two components of each are and . Problem 5exer:GeomInvDistPre <li> Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so. <li> 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. <li> Give a property that is not of interest in Euclidean geometry and is not invariant under distance-preserving maps. Answer <li> 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 . <li> 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. <li> 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.