68Chapter 4(b)IfYis a yaw matrix with yaw angleuthenYT=cosu−sinu0sinucosu0001=cos(−u)sin(−u)0−sin(−u)cos(−u)0001soYTis the matrix representing a yaw transformation with angle−u.It is easily verified thatYTY=Iand hence thatY−1=YT.(c)By the same reasoning used in part (b) you can show that for the pitchmatrixPand roll matrixRtheir inverses are their transposes. So ifQ=Y PRthenQis nonsingular andQ−1= (Y PR)−1=R−1P−1Y−1=RTPTYT14.(b)3/2−2;(c)3/2016.IfL(x) =0for somex=0andAis the standard matrix representation ofL, thenAx=0. It follows from Theorem 1.4.2 thatAis singular.17.The proof is by induction onm. In the case thatm= 1,A1=ArepresentsL1=L. If nowAkis the matrix representingLkand ifxis the coordinatevector ofv, thenAkxis the coordinate vector ofLk(v). SinceLk+1(v) =L(Lk(v))it follows thatAAkx=Ak+1xis the coordinate vector ofLk+1(v).