{VERSION 3 0 "APPLE_PPC_MAC" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 255 1 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 255 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 255 1 0 0 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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 19 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {TEXT 256 47 "Visualizing Linear Transformat ions in Two Space" }}{PARA 258 "" 0 "" {TEXT 257 13 "Tommy Ratliff" }} {PARA 257 "" 0 "" {TEXT 258 18 "September 16, 1998" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 192 "This worksheet contains a procedure LinT ran(A ,L) that will take the list of points L and return a list that \+ contains the image of the points under the linear transformation deter mined by A. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In this sample, I've defined the list " }{TEXT 259 1 "S" } {TEXT 261 1 " " }{TEXT -1 85 "to be the points on the unit square in t he first quadrant, in two space and then let " }{TEXT 260 9 "ImageOfS \+ " }{TEXT -1 126 "be the image of the square under the sheer transforma tion. Then I plot the two lists to see the effect of the transformati on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Be sure to execute each command by pressing return." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "To use this for other tr ansformations, you will just need to change A in step 3 and re-execute 4 and 5." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "1. Define the funct ion LinTran( )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(linalg):" "6#-%%withG6#%'linalgG" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "LinTran := proc(A,L)\n local L2, i; \nL2 := []; \nfor i to nops(L) do \n L2 := [op(L2), convert(multiply(A,L[i]),lis t)];\n od;\nL2;\nend:" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "2. Define S to be the corners of the unit square" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "S := [[0, 0], [1, 0], [1, \+ 1], [0, 1], [0, 0]];" "6#>%\"SG7'7$\"\"!F'7$\"\"\"F'7$\"\"\"\"\"\"7$F' \"\"\"7$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7'7$\"\"!F'7$\" \"\"F'7$F)F)7$F'F)F&" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "3. Defin e A to be the matrix that performs the sheer transformation" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "A := matrix([[1, 1], [0, 1]]);" "6#>%\"AG-% 'matrixG6#7$7$\"\"\"\"\"\"7$\"\"!\"\"\"" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "4. Get the image of the square under the transformation" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "ImageOfS := LinTran(A,S);" "6#>%)Im ageOfSG-%(LinTranG6$%\"AG%\"SG" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 " 5. Plot the result, with S in blue and ImageOfS in red. " }} {PARA 0 "> " 0 "" {XPPEDIT 19 1 "plot([S, ImageOfS],color = [blue, red ],scaling = constrained);" "6#-%%plotG6%7$%\"SG%)ImageOfSG/%&colorG7$% %blueG%$redG/%(scalingG%,constrainedG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 }