Linear Algebra

Topic: Orthonormal Matrices 291( )−ba(ab)(aRep E2 ,E 2(t) =b)−baor to one where it goes a quarter circle counterclockwise.(ab)( )b−a(aRep E2 ,E 2(t) =b)b−aWe can geometrically describe these two cases. Let θ be the counterclockwiseangle between the x-axis and the image of ⃗e 1 . The first matrix above represents,with respect to the standard bases, a rotation of the plane by θ radians.( )−ba(ab)( )x t↦−→y()x cos θ − y sin θx sin θ + y cos θThe second matrix above represents a reflection of the plane through the linebisecting the angle between ⃗e 1 and t(⃗e 1 ).(ab)( )b−a( )x t↦−→y()x cos θ + y sin θx sin θ − y cos θ(This picture shows ⃗e 1 reflected up into the first quadrant and ⃗e 2 reflected downinto the fourth quadrant.)Note: in the domain the angle between ⃗e 1 and ⃗e 2 runs counterclockwise, andin the first map above the angle from t(⃗e 1 ) to t(⃗e 2 ) is also counterclockwise,so it preserves the orientation of the angle. But the second map reverses theorientation. A distance-preserving map is direct if it preserves orientations andopposite if it reverses orientation.So, we have characterized the Euclidean study of congruence. It considers,for plane figures, the properties that are invariant under combinations of (i) arotation followed by a translation, or (ii) a reflection followed by a translation(a reflection followed by a non-trivial translation is a glide reflection).Another idea, besides congruence of figures, encountered in elementarygeometry is that figures are similar if they are congruent after a change of scale.These two triangles are similar since the second is the same shape as the first,but 3/2-ths the size.