{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Exercise" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 -12 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 24 "Linear A lgebra Powertool" }}{PARA 256 "" 0 "" {TEXT -1 32 "Visualizing Linear \+ Transformatio" }{TEXT 256 0 "" }{TEXT -1 6 "ns in " }{XPPEDIT 18 0 "R^ 3;" "6#*$%\"RG\"\"$" }}{PARA 257 "" 0 "" {TEXT -1 10 "OnLine 3.2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Worksheet by Michael K. May, S. J., revised by Russell Blyth" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: \+ with(linalg): with(plots): with(plottools):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Outline" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "This worksheet covers material sim ilar to that covered in the On Line 3.2 section." }}{PARA 0 "" 0 "" {TEXT -1 25 "The basic objectives are:" }}{PARA 14 "" 0 "" {TEXT -1 60 "1) Learn how to have Maple draw a stick figure of a car in " } {XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 50 "2) Learn to represent a linear transformation in " } {XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 32 " as multiplication by a matrix. " }}{PARA 14 "" 0 "" {TEXT -1 97 "3) Explore the relati onships between the matrix of a transformation and its image and nulls pace." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Drawing a 3 dimensiona l car in Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "We first need t o get Maple to draw a 3 dimensional object. We follow the OnLine sect ion in the student resource manual and work with the image of a car. \+ We start by producing a two dimensional outline of a car." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "listcar := [[0,.0954], [.0102,.161 2], [.1425,.2237], [.2124,.2336],\n [.2513, .2401], [.2902, .2401], [.3187, .3158], [.3472, .3553],\n [.3912, .3684], [.4611, .3750], \+ [.5104, .3783], [.5674, .3783],\n [.6373, .3783], [.6710, .3684], [ .6891, .3520], [.7047, .3388],\n [.7202, .3191], [.7306, .2961], [. 7409, .2763], [.7876, .2664],\n [.8135, .2599], [.8264, .2500], [.8 394, .2368], [.8472, .2072],\n [.8497, .1612], [.8497, .1513], [.84 97, .1283], [.8497, .1020],\n [.7350, .1020], [.7350, .0510], [.684 0, .0000], [.6250, .0000],\n [.5740, .0510], [.5740, .1020], [.3410 , .1020], [.3410, .0510],\n [.2900, .0000], [.2310, .0000], [.1800, .0510], [.1800, .1020], \n [.0020, .1020]]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We trace the outline." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "pointplot(listcar, style=line, view=[-0.2..1,0..0.5] , axes=framed, scaling=constrained, color=black, labels=[x,z], labelfo nt=[TIMES, BOLD, 14]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "To tur n the car into a 3 dimensional object, we insert a y coordinate in bet ween the x and z coordinates. We make the car .35 units wide, with th e right side in the x-z plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "listcarright := map(u->[op(1,u), 0, op(2,u)],listcar):\nlistc arleft := map(u->[op(1,u), 0.35, op(2,u)],listcar):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "Let's plot the two sides together, making the \+ right side red and the left side blue. To plot the two sides with spe cified colors we use the display command. Note that the individual pa rts of the graph end with colons." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "rightside:= pointplot3d(listcarright, style=line, co lor=red):\nleftside:= pointplot3d(listcarleft, style=line, color=blue) :\ndisplay(\{rightside,leftside\},axes=normal, scaling=constrained, or ientation=[135,66]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "To make \+ it look like a single object we draw lines connecting the two sides of the car. We do that in green." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "listcarcross := [seq(op([listcarright[k],listcarleft[k]]),k=1. .41)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "crossing:= point plot3d(listcarcross, style=line, color=green):\ndisplay(\{leftside,ri ghtside,crossing\},axes=normal, scaling=constrained, orientation=[135, 66]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Finally, we replace the axes with a garage. We will have transformations act on the car, whi le leaving the garage alone so that it can act as a reference frame. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "garage:=[[0,0,0], [1,0 ,0], [1,.35,0], [1,0,0], [1,0,.5], [1,.35,.5],\n [1,0,.5], [0,0,.5] , [0,.35,.5], [0,0,.5], [0,0,0], [0,.35,0],\n [1,.35,0], [1,.35,.5] , [0,.35,.5], [0,.35,0]]:\ngaragel:= pointplot3d(garage, style=line, \+ color=black):\ndisplay(\{leftside,rightside,crossing,garagel\}, axes=n one, scaling=constrained, orientation=[135,66]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "To reduce typing we define a func tion that turns a list of points into an outline of the desired color " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "poly3d := (pointlist, s hade) -> \n pointplot3d(pointlist, style=line, color=shade):" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This allows us plot the 4 lists t ogether as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "dis play(\n \{poly3d(listcarright,red), poly3d(listcarleft, blue), \n p oly3d(listcarcross, green), poly3d(garage, black)\}, \n axes=none, s caling=constrained, orientation=[135,66]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We use the 4 lists in this example for the remainder of t he worksheet." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 9 "Exercise:" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 138 "1) To check that you understan d the syntax of the commad, plot just the crossings of the car in oran ge together with the garage in green." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Visualizing Linear Transformations in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "In the previous OnLine section we \+ looked at 2 by 2 matrices acting on " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG \"\"#" }{TEXT -1 40 ". We are ready to extend this study to " } {XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 82 ". We borrow a proc edure that allows us to multiply a matrix by a list of vectors." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "multmatbylist := proc(multm at, listofvecs)\n map(x->convert(evalm(multmat&*x),list),listofvecs );\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "We start with a ma trix that rotates objects by 90 degrees in the y-z plane (that is, aro und the x-axis). Such a transformation should send e1 to e1, e2 to e3 , and e3 to -e2. We designate the matrix rot1." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 105 "rot1:= matrix(3,3,[[1,0,0],[0,0,-1],[0,1,0]]) ;\ntestvec := [xcoord, ycoord, zcoord];\nevalm(rot1&*testvec);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Now we want to multiply the three carlists by rot1 and graph the result. Note that we leave the garage alone." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "rlist:=multmatb ylist(rot1,listcarright):\nllist:=multmatbylist(rot1,listcarleft):\ncl ist:=multmatbylist(rot1,listcarcross):\ndisplay(\n \{poly3d(rlist,red ), poly3d(llist, blue), \n poly3d(clist, green), poly3d(garage, blac k)\}, \n axes=none, scaling=constrained, orientation=[135,66]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We see that the t ransformation turns the car on its side. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "As a second example, shear the car in the z direction by the value of x. (This jacks up the back en d of the car.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "shear1:= matrix(3,3,[[1,0,0],[0,1,0],[1,0,1]]);\nrlist2:=multmatbylist(shear1, listcarright):\nllist2:=multmatbylist(shear1,listcarleft):\nclist2:=mu ltmatbylist(shear1,listcarcross):\ndisplay(\n \{poly3d(rlist2,red), p oly3d(llist2, blue), \n poly3d(clist2, green), poly3d(garage, black) \}, \n axes=none, scaling=constrained, orientation=[135,66]);" }}} {EXCHG {PARA 5 "" 0 "" {TEXT -1 10 "Exercises:" }}}{EXCHG {PARA 258 " " 0 "" {TEXT -1 388 "2) Use matrix multiplication to produce a single matrix that composes the rotation in the y-z plane followed by the sh earing in the x-z plane (think carefully about the order of multiplica tion). Compare the result of multiplying this single matrix on the ca r with the result of first multiplying the car by the rotation matrix \+ and then the result of the rotation by the shearing matrix." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 255 "3) Prod uce a matrix to rotate the figure counterclockwise 30 degrees about th e x-axis and another matrix to rotate the image by 20 degrees counterc lockwise about the z axis. Apply these matrices in succession to the \+ image of the car and plot the result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 121 "4) Obtain a single matrix that has the same effect as the composite of the two \+ matrices in Exercise 3. Plot the result." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "5) Write \+ a hypothesis about the result of multiplying the image of the car by a rank 2 matrix. Create a random rank 2 matrix of size 3 by 3 and veri fy your hypothesis. (Your matrix should be of rank 2 with no zeros. \+ Pay particular attention to what happens to the images of the sides of the car.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 416 "6) Write a hypothesis about the result \+ of multiplying the image of the car by a matrix with all zeros in the \+ third column. Create a random 3 by 3 matrix of this type and verify y our hypothesis. (Your matrix should be of rank 2 with zeros only in t he third column. Pay particular attention to what happens to the imag es of the sides of the car. Your hypothesis should mention vectors th at are in the null space.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "7) Write a hypothesis ab out the result of multiplying the image of the car by a rank 1 matrix. Create a random rank 1 matrix of size 3 by 3 and verify your hypothe sis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 271 "8) Write a hypothesis about the result of mul tiplying the image of the car by a matrix with all zeros in the first \+ and third rows. Create a random 3 by 3 matrix of this type and verify your hypothesis. (Your hypothesis should mention interesting vectors in the image.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }