Linear Algebra

286 Chapter 3. Maps Between Spaces
Topic: Orthonormal Matrices
In The Elements, Euclid considers two figures to be the same if they have the
same size and shape. That is, the triangles below are not equal because they are
not the same set of points. But they are congruent—essentially indistinguishable
for Euclid’s purposes—because we can imagine picking up the plane up, sliding
it over and turning it a bit (although not bending it or stretching it), and then
putting it back down, to superimpose the first figure on the second.
P1
P2
P3
(Euclid never explicitly states this principle but he uses it often [Casey].) In
modern terms, “picking the plane up ... ” means taking a map from the plane
to itself. We, and Euclid, are considering only certain transformations of the
plane, ones that may possibly slide or turn the plane but not bend or stretch
it. Accordingly, we define a function f : R 2 → R 2 to be distance-preserving (or
a rigid motion, orisometry) if for all points P1,P2 ∈ R 2 , the map satisfies the
condition that the distance from f(P1) tof(P2) equals the distance from P1 to
P2. We define a plane figure to be a set of points in the plane and we say that
two figures are congruent if there is a distance-preserving map from the plane
to itself that carries one figure onto the other.
Many statements from Euclidean geometry follow easily from these definitions.
Some are: (i) collinearity is invariant under any distance-preserving map
(that is, if P1, P2, andP3 are collinear then so are f(P1), f(P2), and f(P3)),
(ii) betweeness is invariant under any distance-preserving map (if P2 is between
P1 and P3 then so is f(P2) between f(P1) andf(P3)), (iii) the property of
being a triangle is invariant under any distance-preserving map (if a figure is a
triangle then the image of that figure is also a triangle), (iv) and the property of
being a circle is invariant under any distance-preserving map. In 1872, F. Klein
suggested that Euclidean geometry can be characterized as the study of properties
that are invariant under distance-preserving maps. (This forms part of
Klein’s Erlanger Program, which proposes the organizing principle that each
kind of geometry—Euclidean, projective, etc.—can be described as the study
of the properties that are invariant under some group of transformations. The
word ‘group’ here means more than just ‘collection’, but that lies outside of our
scope.)
We can use linear algebra to characterize the distance-preserving maps of
the plane.
First, there are distance-preserving transformations of the plane that are not
linear. The obvious example is this translation.
� � � � � �
x
x 1
↦→ + =
y
y 0
Q1
Q2
Q3
� �
x +1
y
However, this example turns out to be the only example, in the sense that if f
is distance-preserving and sends �0 to�v0 then the map �v ↦→ f(�v) − �v0 is linear.