Linear Algebra

284 Chapter Three. Maps Between SpacesWe can geometrically describe these two cases. Let θ be the angle betweenthe x-axis and the image of ⃗e 1 , measured counterclockwise. The first matrixabove represents, with respect to the standard bases, a rotation of the plane byθ radians.( )−ba(ab)(xy)t↦−→( )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(xy)t↦−→( )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 again: the angle between ⃗e 1 and ⃗e 2 runs counterclockwise, and in thefirst map above the angle from t(⃗e 1 ) to t(⃗e 2 ) is also counterclockwise, so theorientation of the angle is preserved. But in the second map the orientation isreversed. 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 elementary geometryis 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.P 1P 2P 3Q 1Q 2Q 3From the above work, we have that figures are similar if there is an orthonormalmatrix T such that the points ⃗q on one are derived from the points ⃗p by ⃗q =(kT )⃗v + ⃗p 0 for some nonzero real number k and constant vector ⃗p 0 .