Linear Algebra

TopicOrthonormal MatricesIn The Elements, Euclid considers two figures to be the same if they havethe same size and shape. That is, while the triangles below are not equalbecause they are not the same set of points, they are congruent — essentiallyindistinguishable for Euclid’s purposes — because we can imagine picking theplane up, sliding it over and rotating it a bit, although not warping or stretchingit, and then putting it back down, to superimpose the first figure on the second.(Euclid never explicitly states this principle but he uses it often [Casey].)P 1P 2P 3Q 1Q 2Q 3In modern terminology, “picking the plane up . . . ” is considering a map fromthe plane to itself. Euclid considers only transformations of the plane that mayslide or turn the plane but not bend or stretch it. Accordingly, we define a mapf: R 2 → R 2 to be distance-preserving or a rigid motion or an isometry, if forall points P 1 , P 2 ∈ R 2 , the distance from f(P 1 ) to f(P 2 ) equals the distance fromP 1 to P 2 . We also define a plane figure to be a set of points in the plane and wesay that two figures are congruent if there is a distance-preserving map fromthe 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 P 1 , P 2 , and P 3 are collinear then so are f(P 1 ), f(P 2 ), and f(P 3 )), (ii) betweenessis invariant under any distance-preserving map (if P 2 is between P 1 and P 3 thenso is f(P 2 ) between f(P 1 ) and f(P 3 )), (iii) the property of being a triangle isinvariant under any distance-preserving map (if a figure is a triangle then theimage of that figure is also a triangle), (iv) and the property of being a circleis invariant under any distance-preserving map. In 1872, F. Klein suggestedthat we can define Euclidean geometry as the study of properties that areinvariant under these maps. (This forms part of Klein’s Erlanger Program,which proposes the organizing principle that we can describe each kind ofgeometry — Euclidean, projective, etc. — as the study of the properties that are