{VERSION 5 0 "SUN SPARC SOLARIS" "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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 114 " Math 2270\nMaple Projec t 2\nOctober 14, 20002. \n\nA Maple text version of this project may b e found at the web site" }}{PARA 0 "" 0 "" {TEXT -1 51 "http://www.mat h.utah.edu/~kapovich/teaching11.html/" }}{PARA 0 "" 0 "" {TEXT -1 59 " Go to that page and click on the \"2-nd maple assignment\". " }} {PARA 0 "" 0 "" {TEXT -1 78 "I recommend saving the file from the webs ite onto you home directory, and then" }}{PARA 0 "" 0 "" {TEXT -1 69 " opening it from Maple, as a ``Maple text'' document. You can then do " }}{PARA 0 "" 0 "" {TEXT -1 69 "the assignment by inserting the prope r commands as well as italicized" }}{PARA 0 "" 0 "" {TEXT -1 65 "textu al comments in answer to the various questions. In order to" }}{PARA 0 "" 0 "" {TEXT -1 69 "execute multiline commands from the version you get from the web, you" }}{PARA 0 "" 0 "" {TEXT -1 63 "should use the \+ ``Edit'' option in your maple window to ``join''" }}{PARA 0 "" 0 "" {TEXT -1 62 "execution groups which you have highlighted with your mou se. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " I remind that in each problem you should use Maple " }}{PARA 0 "" 0 " " {TEXT -1 54 "to solve the problem, not pen-and-paper computations! \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "PART \+ 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "I n the first part of the assignment we will study the geometric meaning of linear (hence matrix) maps" }}{PARA 0 "" 0 "" {TEXT -1 105 "f(x)=A x, from R^n to R^m. We will focus on maps from R^2 to R^2 to illustra te more general properties. " }}{PARA 0 "" 0 "" {TEXT -1 76 "Of cours e, computer graphics are most concerned with those maps and with R^3" }}{PARA 0 "" 0 "" {TEXT -1 52 "rotation maps and projection maps from \+ R^3 to R^2. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 67 " The very definition of a linear map, namely that f(u+v)= f(u)+f(v) " }}{PARA 0 "" 0 "" {TEXT -1 70 " and f(su)=sf(u), for all v ectors u,v and scalars s, has the following" }}{PARA 0 "" 0 "" {TEXT -1 25 " geometric consequences:\n" }}{PARA 0 "" 0 "" {TEXT -1 60 " Lin es are mapped to lines, and parallel lines are mapped to" }}{PARA 0 " " 0 "" {TEXT -1 66 "parallel lines: This is because a line can be des cribed as a set " }}{PARA 0 "" 0 "" {TEXT -1 74 "L=\{u + t v , where \+ t is in R\}, where u is a point on the line and v is " }}{PARA 0 "" 0 "" {TEXT -1 77 "a direction vector. Therefore the image set f(L): =\{f(u+tv)\}=\{f(u)+tf(v)\} " }}{PARA 0 "" 0 "" {TEXT -1 75 "is also \+ a line, going through f(u) with direction f(v). (If the directions" } }{PARA 0 "" 0 "" {TEXT -1 66 "f(v) turns out to be 0, then the line de generates into a point.) \n" }}{PARA 0 "" 0 "" {TEXT -1 67 "Therefore, line segments are mapped to line segments, polygons are" }}{PARA 0 " " 0 "" {TEXT -1 65 "mapped to polygons, and regions bounded by polygon s are mapped to" }}{PARA 0 "" 0 "" {TEXT -1 70 "regions bounded by pol ygons; if we know where the vertices go, we know" }}{PARA 0 "" 0 "" {TEXT -1 61 "everything. Let's use these facts to draw the images of \+ some" }}{PARA 0 "" 0 "" {TEXT -1 38 "polygonal regions under a matrix \+ map.\n" }}{PARA 0 "" 0 "" {TEXT -1 57 " First load the linear algeb ra and plotting libraries\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with (plots):with(linalg):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " verts0:=[[0,0],[1,0],[1,1],[0,1]]; \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " #corners of unit square\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "unitsquare:=polygonplot(verts0, color=`yellow `):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " #this comman d make a polygon and colors\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " #the region inside it yellow. Make sure\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " #to end this command with a c olon!\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(unitsqua re); #Now semicolon! this command\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " #shows the square\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "When you executed the sequence of commands above you shou ld have" }}{PARA 0 "" 0 "" {TEXT -1 67 "gotten a picture of a yellow u nit square. Now we will use a linear" }}{PARA 0 "" 0 "" {TEXT -1 70 " map with matrix A defined below to map this square to a parallelegram \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=matrix([[3,2],[-1,1]]); # the matrix of our\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " \+ #random linear transformation\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->evalm(A&*x); #our linear map\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 10 "Problem 1." }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "1a) " }{TEXT 287 11 "Assignment:" }{TEXT -1 63 " For our map f(x)=Ax defined above, wh at are the images of the" }}{PARA 0 "" 0 "" {TEXT -1 68 "points e1=[1, 0] and e2=[0,1]? How do you find these images from the" }}{PARA 0 "" 0 "" {TEXT -1 22 "rows or columns of A?\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We use the ``map'' command below to see where the vertice s of the unit" }}{PARA 0 "" 0 "" {TEXT -1 70 "square are sent by f. T he syntax is to put the mapping function in as" }}{PARA 0 "" 0 "" {TEXT -1 62 "the first argument, and the list of input points as the s econd" }}{PARA 0 "" 0 "" {TEXT -1 63 "argument. The result of the comm and will be the list of output" }}{PARA 0 "" 0 "" {TEXT -1 66 "points . Check (not to hand in) that this is what happens below to" }}{PARA 0 "" 0 "" {TEXT -1 37 "the four points in the list verts0. \n" }} {PARA 0 "" 0 "" {TEXT -1 70 "Note that here we are conflating points a nd vectors in R^2, namely we " }}{PARA 0 "" 0 "" {TEXT -1 66 "identify each point P with its coordinates, i.e. with the vector" }}{PARA 0 "" 0 "" {TEXT -1 5 " -->" }}{PARA 0 "" 0 "" {TEXT -1 5 " OP\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "verts1:=map(f,verts0);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "image1:=polygonplot(verts1, color=`red`):\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "display(\{unitsquare,image1\});\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "1b) " }{TEXT 286 11 "Assignment: " }{TEXT -1 66 " Explain where f(e1) and f(e2) are represented in the \+ picture you " }}{PARA 0 "" 0 "" {TEXT -1 49 "just made. (Mark them by hand on the picture.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "By the way, if you click on the plot you can choos e to" }}{PARA 0 "" 0 "" {TEXT -1 70 "have the projection ``constrained '', in which case Maple will use the" }}{PARA 0 "" 0 "" {TEXT -1 70 " same scales for the vertical and horizontal axis. Otherwise it scales " }}{PARA 0 "" 0 "" {TEXT -1 66 "the x and y-axes to make the picture \+ fit nicely onto your screen.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 " 1c) " }{TEXT 285 11 "Assignment:" }{TEXT -1 64 " If we compute f(f(x) ):=f^2(x), then what will the matrix be for" }}{PARA 0 "" 0 "" {TEXT -1 67 "the resulting linear transformation? After answering that quest ion," }}{PARA 0 "" 0 "" {TEXT -1 64 "make the vertices for f(f(unitsqu are)) as follows, and draw the " }}{PARA 0 "" 0 "" {TEXT -1 30 "corres ponding image polygons.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "verts2:=map(f,verts1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "image2:=polygonplot(verts2,`color`=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(\{unitsquare,image1,image2\});" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "1d) " }{TEXT 284 11 "Assignment:" }{TEXT -1 63 " Explain what the colum ns of the matrix for f^2 have to do with" }}{PARA 0 "" 0 "" {TEXT -1 33 "the picture you just made above.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "1e) " }{TEXT 283 11 "Assignment:" }{TEXT -1 68 " What is the inverse function to f ? (Hint : what is its matrix? ) " }}{PARA 0 "" 0 "" {TEXT -1 88 "Verify that the inverse mapping takes image1 ba ck to the unit square by using the method" }}{PARA 0 "" 0 "" {TEXT -1 11 "from 1c). \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 276 10 "Problem 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 " Translations of objects are mapped to tr anslations of the image " }}{PARA 0 "" 0 "" {TEXT -1 67 "objects and s calings of objects are mapped to scalings of the image" }}{PARA 0 "" 0 "" {TEXT -1 11 "objects: \n" }}{PARA 0 "" 0 "" {TEXT -1 60 " Fir st let's review what do we mean by an ``object'' and" }}{PARA 0 "" 0 "" {TEXT -1 71 "by translations and scalings of an object. An object \+ is some set S " }}{PARA 0 "" 0 "" {TEXT -1 70 "of points s. By t he image of S we mean the collection of image " }}{PARA 0 "" 0 "" {TEXT -1 75 "points f(s) . We write f(S) for this image (like we did for the line " }}{PARA 0 "" 0 "" {TEXT -1 75 "L and its image f( L) in problem 1). Similarly if b is a translation" }}{PARA 0 " " 0 "" {TEXT -1 73 "vector, then the object S+b means all points of \+ the form s+b where s" }}{PARA 0 "" 0 "" {TEXT -1 74 "is in S. If \+ c is a scalar, then the scaled (or dilated) set cS means" }}{PARA 0 "" 0 "" {TEXT -1 49 "all points of the form cs , where s is in S .\n" }}{PARA 0 "" 0 "" {TEXT -1 69 " Now, if we apply the linear ma p f to the translated object S+b we" }}{PARA 0 "" 0 "" {TEXT -1 80 "ge t all points of the form f(s+b)=f(s)+ f(b), where s is in S (since \+ f is " }}{PARA 0 "" 0 "" {TEXT -1 71 "linear), i.e. exactly the set f(S)+f(b) , which is the translation of " }}{PARA 0 "" 0 "" {TEXT -1 73 "the image f(S) by the vector f(b). Similarly, if we apply \+ f to the " }}{PARA 0 "" 0 "" {TEXT -1 70 "scaled set cS we get f(cS ) to be the set of all points f(cs)=cf(s) " }}{PARA 0 "" 0 "" {TEXT -1 69 "(since f is linear), i.e. the scaling cf(S) of the image set \+ f(S).\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 " Here's how to tra nslate and scale the unit square from #1: " }}{PARA 0 "" 0 "" {TEXT -1 42 "Let's translate it by the vector b=[2,3]:\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "b:=[2,3];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "trans:=x->evalm(x+b);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "verts3:=map(trans,verts0);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "image3:=polygonplot(verts3,color=`yellow`):\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " #colon!\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(unitsquare,image3);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Here's how to scale it by a factor of 0.2:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c:=0.2;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "shrink:=x->evalm(c*x);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "verts4:=map(shrink,verts0); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "image4:=polygonplot(v erts4,color=`red`):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "di splay(\{image4,unitsquare\});\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "2a ) " }{TEXT 281 11 "Assignment:" }{TEXT -1 61 " Describe where you exp ect the translated unit square, image3" }}{PARA 0 "" 0 "" {TEXT -1 69 "above, to be mapped by our linear map f from problem 1. Then use " }}{PARA 0 "" 0 "" {TEXT -1 69 "the ``map'' command and the ``trans'' command, as well as polygonplot" }}{PARA 0 "" 0 "" {TEXT -1 67 "and d isplay, to make a picture of the images of the unit square and" }} {PARA 0 "" 0 "" {TEXT -1 68 "its translation when they are mapped by \+ f . Verify that the images" }}{PARA 0 "" 0 "" {TEXT -1 36 "differ by \+ the expected translation.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 6 "2b) " }{TEXT 282 11 "Assignment:" } {TEXT -1 61 " Describe what you expect the image of the scaled down sq uare" }}{PARA 0 "" 0 "" {TEXT -1 90 "above to be when you apply f , a nd then draw (using MAPLE!) a picture illustrating this.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 10 "Problem \+ 3." }{TEXT -1 88 " Special linear transformations in R^2 (rotations, reflections, projections, shears):\n" }}{PARA 0 "" 0 "" {TEXT -1 62 " 3a) Rotations: The matrix for rotating by an angle theta is:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Rot:=theta->matrix([[cos(theta),-si n(theta)],[sin(theta),cos(theta)]]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "so to rotate the vector (2,3) by Pi/3, we would command\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "theta:=Pi/3;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(Rot(theta)&*[2,3]);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 279 18 "Assignment for 3a)" }{TEXT -1 61 ": Draw a picture of \+ the unit square rotated by Pi/3 radians.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "3b) Reflections (see section 2.2 of the textbook)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 11 "Assignme nt:" }{TEXT -1 53 " Define matrices which give reflections across th e " }}{PARA 0 "" 0 "" {TEXT -1 57 "x-axis, across the line y=-2x, and \+ across the line y=x. " }}{PARA 0 "" 0 "" {TEXT -1 86 "Illustrate what each of these linear maps does to the unit square (as we did in 1a). \+ " }}{PARA 0 "" 0 "" {TEXT -1 86 "Finally, verify (using MAPLE computa tion) that reflecting across the line y=x is the " }}{PARA 0 "" 0 "" {TEXT -1 90 "composition of first rotating by Pi/4 in the clockwise \+ direction, then reflecting across" }}{PARA 0 "" 0 "" {TEXT -1 63 "the \+ x-axis, then rotating (back ) Pi/4 in the counterclockwise " }}{PARA 0 "" 0 "" {TEXT -1 11 "direction. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Hint: if L: x --> y and M: y-- > z are linear transformations " }}{PARA 0 "" 0 "" {TEXT -1 70 "and N: x--> y=L(x) --> z= M(y) is their composition, and the matrix " }} {PARA 0 "" 0 "" {TEXT -1 74 "of L is A, the matrix for M is B, t hen the matrix for N is A&*B. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "3c) You can scale by different factors in the x and y-directions. " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 288 11 "Assignment:" }{TEXT -1 55 " define a matrix expands \+ the x-direction by 3 and the " }}{PARA 0 "" 0 "" {TEXT -1 99 "y-direct ion by 2? (This is an analogue of the \"Procustes\" transformation dis cussed in the class). " }}{PARA 0 "" 0 "" {TEXT -1 86 "Make a picture of what this transformation does to the unit square similarly to 1a). \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "3d) Projections. " } {TEXT 289 11 "Assignment:" }{TEXT -1 50 " Write down the matrix which \+ projects points onto " }}{PARA 0 "" 0 "" {TEXT -1 68 "the line y=2x. \+ Illustrate what this projection map does to the unit" }}{PARA 0 "" 0 " " {TEXT -1 35 "square. Is there an inverse map? \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "3e) Shears: Recall that a shear of strength \+ k in the x-direction is defined by the matrix\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "Shear:=k->matrix([[1,k],[0,1]]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 290 11 "Assignment:" }{TEXT -1 66 " For k=1 explore what the shear map does to the unit square. What \+ " }}{PARA 0 "" 0 "" {TEXT -1 84 "happens if you apply the shear map tw ice? Three times? Make pictures using MAPLE. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "PART 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "This part of the project is about the fundamental subs paces associated with" }}{PARA 0 "" 0 "" {TEXT -1 64 "matrices and abo ut coordinates with respect to different bases. " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Here's the matr ix we will work with. We will keep it moderately" }}{PARA 0 "" 0 "" {TEXT -1 65 "sized, but of course we could make it much bigger without causing" }}{PARA 0 "" 0 "" {TEXT -1 20 "Maple any problems. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A:= matrix([[1,1,1,2,6],[2,3,-2,1,-3],[3,5,-5,1,-8],[4,3,8,2,3]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"F*F*\"\"# \"\"'7'F+\"\"$!\"#F*!\"$7'F.\"\"&!\"&F*!\")7'\"\"%F.\"\")F+F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b:=vector([0,0,0,0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7&\"\"!F)F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "We will think of this as the matri x of a linear transformation " }{MPLTEXT 1 0 12 "L: R^5-> R^4" }{TEXT -1 1 "," }{MPLTEXT 1 0 10 " L(x)=Ax." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 11 "Problem 4. " }{TEXT -1 1 " " }{TEXT 291 11 "Assignment:" }{TEXT -1 48 " Compute rank and n ullity of A using reduction " }}{PARA 0 "" 0 "" {TEXT -1 26 "of A to \+ row echelon form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 10 "Problem \+ 5." }{TEXT -1 2 " " }{TEXT 292 11 "Assignment:" }{TEXT -1 35 " Find a basis for the kernel of A. " }}{PARA 0 "" 0 "" {TEXT -1 76 "To solve \+ this problem you can either use the \"rref\" command or \"linsolve\": \+ \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "linsolve(A, b);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "In this command A is the matrix , b is the vector \nof the right-hand side (in the present problem y ou will use \"zero\" vector)." }}{PARA 0 "" 0 "" {TEXT -1 93 "If you d o not want to use the symbols like _t_1 as your parameters you can us e the command:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "linsolv e(A, b, 'r', s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now the names of the parameters will be s_1, s_2,... Type " }}{PARA 0 "" 0 "" {TEXT -1 3 ">r;" }}{PARA 0 "" 0 "" {TEXT -1 39 "to see what the rank o f the matrix is. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 263 10 "Problem 6." }{TEXT -1 2 " " }{TEXT 293 11 "Assig nment:" }{TEXT -1 64 " Use Maple to find a basis v1, v2, v3 for the co lumn space of A " }}{PARA 0 "" 0 "" {TEXT -1 75 "(i.e. of the image of L). Recall that one way to find this basis is to do" }}{PARA 0 "" 0 "" {TEXT -1 63 "row operations, putting your matrix into reduced col umn echelon" }}{PARA 0 "" 0 "" {TEXT -1 7 "form. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 11 "Problem \+ 7. " }{TEXT -1 66 " Maple will compute bases for these spaces with si ngle commands. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 295 11 "Assig nment:" }{TEXT -1 69 " Compare the answers you've gotten above with Ma ple's answers: Since" }}{PARA 0 "" 0 "" {TEXT -1 70 "you know bases a re not unique, you can pretty well guess that Maple is" }}{PARA 0 "" 0 "" {TEXT -1 68 "using methods close to the ones we used, since its a nswers should be" }}{PARA 0 "" 0 "" {TEXT -1 31 "quite similar to some of yours:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "rowspace(A); # Gives a basis for the space spanned \+ by row vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nullspac e(A);#nullspace basis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "co lspace(A); #nice column space basis, i.e. a basis for the image of L. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v:=convert(colspace(A), list); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "#The above command is \+ making a list of vectors in the basis. This keeps track of order in th e set " }{TEXT 268 1 "v" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "#of vectors which form basis of the image of L. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " v1:=v[1]; v2:= v[2]; v3:= v[3] " }}{PARA 0 " " 0 "" {TEXT -1 102 "These vectors form a basis of the image of L. Com pare these vectors with vectors from the problem 6. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "r1:=row(rref(A),1); #r_1,r_2,r_3 basis for rowsp ace(A)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r2:=row(rref(A),2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r3:=row(rref(A),3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G:=convert(nullspace(A),l ist); #making a list" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " \+ #keeps track of order in a set" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n1:=G[1];n2:=G[2]; #nullspace basis. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "Proble m 8. " }{TEXT -1 1 " " }{TEXT 294 11 "Assignment:" }{TEXT -1 78 " Use \+ Maple to verify that the vectors r1, r2, r3, n1, n2 form a basis of R^ 5. " }}{PARA 0 "" 0 "" {TEXT 269 5 "Hint:" }{TEXT -1 50 " use reduced \+ row echelon form or the determinant. " }}{PARA 0 "" 0 "" {TEXT 270 21 "Explain your solution" }{TEXT -1 83 ": namely, how the value of the d eterminant or shape of the reduced echelon form is " }}{PARA 0 "" 0 " " {TEXT -1 128 "related to the fact that we have a basis. Check that e ach of the vectors r1, r2, r3 is orthogonal to each of the vectors n1, n2. " }}{PARA 0 "" 0 "" {TEXT -1 64 "I.e. the row-space is orthogonal to the kernel of the matrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 261 10 "Problem 9." }{TEXT -1 3 " " } {TEXT 296 11 "Assignment:" }{TEXT -1 63 " Let us call E=\{e1,e2,e3,e4, e5\}, i.e. the set of standard basis" }}{PARA 0 "" 0 "" {TEXT -1 64 "v ectors in R^5. Let us call our new basis S=\{r1,r2,r3,n1,n2\}. " }} {PARA 0 "" 0 "" {TEXT -1 29 "Find the transition matrices " }{MPLTEXT 1 0 9 "P_\{E <-S\}" }{TEXT -1 4 " and" }{TEXT 267 1 " " }{MPLTEXT 1 0 8 "P_\{S<-E\}" }{TEXT -1 8 ", which " }}{PARA 0 "" 0 "" {TEXT -1 78 "c onvert S coordinates to E-coordinates and vice-versa. Recall that the \+ matrix " }{MPLTEXT 1 0 9 "P_\{E <-S\}" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "is very easy to find and the matrix " }{MPLTEXT 1 0 8 "P_\{S<-E\}" }{TEXT -1 36 " is easily obtained from the matrix " } {MPLTEXT 1 0 12 "P_\{E <-S\}. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 271 12 "Problem 10. " }{TEXT -1 85 "Using transition matrix find the coordinates of the following vectors with \+ respect to" }}{PARA 0 "" 0 "" {TEXT -1 12 "the S-basis:" }}{PARA 0 "" 0 "" {TEXT 259 4 "10a)" }{TEXT -1 18 " a=(0,1,-4,0,-3)," }}{PARA 0 " " 0 "" {TEXT 258 4 "10b)" }{TEXT -1 16 " b=(1,0,0,0,0)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}}{MARK "66 3 0" 99 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }