Linear Algebra, 2020a

324 Chapter Three. Maps Between Spaces
We could also make the cube appear to be moving away from us by producing
film frames of it shrinking, which gives us figures that are similar.
Frame 1: Frame 2: Frame 3:
Computer graphics incorporates techniques from linear algebra in many other
ways (see Exercise 4).
A beautiful book that explores some of this area is [Weyl]. More on groups,
of transformations and otherwise, is in any book on Modern Algebra, for instance
[Birkhoff & MacLane]. More on Klein and the Erlanger Program is in [Yaglom].
Exercises
1 Decide ( if√each of these √ ) is an orthonormal matrix.
1/ 2 −1/ 2
(a)
−1/ √ 2 −1/ √ 2
( √ √ )
1/ 3 −1/ 3
(b)
−1/ √ 3 −1/ √ 3
( √ √ √ )
1/ 3 − 2/ 3
(c)
− √ 2/ √ 3 −1/ √ 3
2 Write down the formula for each of these distance-preserving maps.
(a) the map that rotates π/6 radians, and then translates by ⃗e 2
(b) the map that reflects about the line y = 2x
(c) the map that reflects about y =−2x and translates over 1 and up 1
3 (a) 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.
(b) Prove that congruence is an equivalence relation between plane figures.
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.
⎛ ⎞ ⎛ ⎞
( ) ( a c e x
a c x e
+ ⎝ ⎠ ⎝ ⎠
b d)(
y f)
b d f
0 0 1
(These are homogeneous coordinates; see the Topic on Projective Geometry).
5 (a) Verify that the properties described in the second paragraph of this Topic as
invariant under distance-preserving maps are indeed so.
(b) Give two more properties that are of interest in Euclidean geometry from
your experience in studying that subject that are also invariant under distancepreserving
maps.
(c) Give a property that is not of interest in Euclidean geometry and is not
invariant under distance-preserving maps.
y
1