1 1 1 1 2 1 2 1 1 2 1 2 d rp 1 1 1 1 2 1 2 1 1 2 1 2

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100001010=1212000112120(d)RP=1000010101201201012012=1201212012010(e)Y PR=12120121200010010101001000121201212=001010100(f)RPY=10001212012120010101001212012120001=00110001012. (a)IfYis the yaw matrix and we expand det(Y) along its third row we getdet(Y) = cos2u+ sin2u= 1Similarly, if we expand the determinant pitch matrixPalong its secondand expand the determinant of the roll matrixRalong its first row wegetdet(P) = cos2v+ sin2v= 1det(R) = cos2w+ sin2w= 1
68Chapter 4(b)IfYis a yaw matrix with yaw angleuthenYT=cosusinu0sinucosu0001=cos(u)sin(u)0sin(u)cos(u)0001soYTis the matrix representing a yaw transformation with angleu.It is easily verified thatYTY=Iand hence thatY1=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 andQ1= (Y PR)1=R1P1Y1=RTPTYT14.(b)3/22;(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).