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{SECT 0 {PARA 270 "" 0 "" {TEXT -1 0 "" }{TEXT 295 22 "Linear Transfor
mations" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT -1 18 "Jonathan Spingarn " }}{PARA 259 "" 0 "" {TEXT -1
13 "Chad Mullikin" }}{PARA 260 "" 0 "" {TEXT -1 21 "School of Mathemat
ics" }}{PARA 261 "" 0 "" {TEXT -1 31 "Georgia Institute of Technology
" }}{PARA 269 "" 0 "" {TEXT -1 11 "Winter 1999" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 40 " The purpose of this worksheet is to" }{TEXT -1 226 " famil
iarize the student with the geometric properties of a linear transform
ation. With these properties understood it should then be less difficu
lt to understand some of the more general properties of linear transfo
rmations. " }}{PARA 0 "" 0 "" {TEXT -1 305 " This worksheet contain
s several algorithms which calculate a transformation matrix for the s
tudent. Followed by examples on how to use each algorithm and then exe
rcises related to the transformations. It should be emphasized that th
e goal of these exercises is not to teach a student how to use Maple.
" }}{PARA 0 "" 0 "" {TEXT -1 378 " To begin, open the first block \+
of minimized code marked \"Procedures\" and execute the block. To do t
his simply click on the \"+\" symbol and then put the cursor at the be
ginning of any line in the execution block and hit the enter key. Then
the block may be minimized and ignored. However, for those interested
in the mechanics of this worksheet feel free to examine the code." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 257 "" 0 "" {TEXT 269
10 "Procedures" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 0 "" }
{TEXT 263 67 "This code will need to be executed once and you may then
ignore it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 28 "Last:=matrix([[1,0],[0,1]]):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "Id:=matrix([[1,0],[0,1]]):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 29 "Total:=matrix([[1,0],[0,1]]):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 21 "current:=array(0..6):" }}{PARA 0 "> " 0 "" {MPLTEXT
1 0 21 "initial:=array(0..6):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "in
itial[0]:=([-1,1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "initial[1]:=
([-1,-1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "initial[2]:=([1,-1]):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "initial[3]:=([1,1]):" }}{PARA
262 "> " 0 "" {MPLTEXT 1 0 20 "initial[4]:=([2,0]):" }}{PARA 263 "> "
0 "" {MPLTEXT 1 0 21 "initial[5]:=([1,-1]):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 21 "initial[6]:=([-1,1]):" }}{PARA 264 "> " 0 ""
{MPLTEXT 1 0 802 "makeplot := proc (vecs)\n local plotarray,i,ip;\n \+
plotarray:=array(0..8);\n for i from 0 to 6 do \n ip:=(i+1) mod 7;\n
plotarray[i]:=plot([(1-t)*vecs[i][1]+t*vecs[ip][1], \+
(1-t)*vecs[i][2]+t*vecs[ip][2],t=0..1],\n -6..6,-6..6,scaling=con
strained);\n od:\nplotarray[7]:=plot([(1-t)*vecs[0][1]+t*vecs[3][1], \+
(1-t)*vecs[0][2]+t*vecs[3][2],t=0..1],\n -6..6,
-6..6,color=blue,scaling=constrained);\nplotarray[8]:=plot([(1-t)*vecs
[3][1]+t*vecs[1][1], (1-t)*vecs[3][2]+t*vecs[1][2]
,t=0..1],\n -6..6,-6..6,scaling=constrained);\nprint(\"Last\"), pri
nt(Last), printf(\"\\n\"),print(\"Total\"), print(Total): dis
play(\{plotarray[0], plotarray[1],plotarray[2], plotarray[3], plotarra
y[4], plotarray[5], plotarray[6], plotarray[7], plotarray[8]\}):\nend:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "reset:=proc()\n local i;\n g
lobal current, initial, Total, Last, Id;\n for i from 0 to 6 do\n cur
rent[i]:=initial[i];\n od;\nLast:=Id;\nTotal:=Id;\nmakeplot(current):
\nend: " }}{PARA 256 "> " 0 "" {MPLTEXT 1 0 276 "rotate := proc(angle)
\n local s,c,i;\n global Last, Total, current;\n s:=evalf(sin(angle
)); c:=evalf(cos(angle));\n Last:=matrix([[c,-s],[s,c]]);\n for i fr
om 0 to 6 do\n current[i]:=evalm(Last &* current[i]);\n od:\n Tot
al:=evalm(Last &* Total):\n makeplot(current):\n end:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 271 "project:=proc(a,b)\n local i,dot;\n global \+
Last, Total, current;\n dot:=a^2+b^2;\n Last:=matrix([[a^2/dot,a*b/d
ot],\n [b*a/dot,b^2/dot]]);\n for i from 0 to 6 do\n current[i]:
=evalm(Last &* current[i]);\n od:\n Total:=evalm(Last &* Total):\n \+
makeplot(current): \nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "ref
lect:=proc(a,b)\n local i,dot;\n global Last, Total, current;\n dot
:=a^2+b^2;\n Last:=matrix([[(2*a^2/dot)-1,2*a*b/dot],\n [2*b*a/dot,
(2*b^2/dot)-1]]);\n for i from 0 to 6 do\n current[i]:=evalm(Last \+
&* current[i]);\n od:\n Total:=evalm(Last &* Total):\n makeplot(cur
rent): \nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "mat_column:=pro
c(a,b,c,d)\n local i;\n global Last, Total, current;\n Last:=matrix
([[a,c],\n [b,d]]);\n for i from 0 to 6 do\n current[i
]:=evalm(Last &* current[i]);\n od:\n Total:=evalm(Last &* Total):\n
makeplot(current): \nend:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "mat_row:=proc(a,b,c,d)\n local i;
\n global Last, Total, current;\n Last:=matrix([[a,b],\n \+
[c,d]]);\n for i from 0 to 6 do\n current[i]:=evalm(Last &* curre
nt[i]);\n od:\n Total:=evalm(Last &* Total):\n makeplot(current): \+
\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "reset();" }}{PARA 7 "" 1
"" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 ""
{TEXT -1 33 "Warning, new definition for trace" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 264 6 "Synta
x" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The operations defined for yo
u in this worksheet are as follows." }{TEXT 261 0 "" }{TEXT 260 0 "" }
}{PARA 0 "" 0 "" {TEXT 296 14 "rotate(angle);" }}{PARA 0 "" 0 ""
{TEXT -1 109 "rotates through the input angle (in Radians) with respec
t to the x-axis in the counter clockwise direction\n" }{TEXT 256 13
"project(a,b);" }{TEXT 259 1 "\n" }{TEXT -1 48 "projection onto the li
ne through (0,0) and (a,b)" }}{PARA 0 "" 0 "" {TEXT 257 13 "reflect(a,
b);" }{TEXT 265 1 "\n" }{TEXT -1 51 "reflection across the line throug
h (0,0) and (a,b)\n" }{TEXT 297 17 "mat_row(a,b,c,d);" }{TEXT -1 53 " \+
\napplies the linear transformation whose matrix is\n" }{XPPEDIT 18
0 "matrix([[a, b], [c, d]]);" "6#-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"d
G" }{TEXT -1 1 "\n" }{TEXT 256 20 "mat_column(a,b,c,d);" }{TEXT 258 1
"\n" }{TEXT -1 50 "applies the linear transformation whose matrix is\n
" }{XPPEDIT 18 0 "matrix([[a, c], [b, d]]);" "6#-%'matrixG6#7$7$%\"aG%
\"cG7$%\"bG%\"dG" }{TEXT -1 1 "\n" }{TEXT 266 8 "reset();" }{TEXT -1
57 "\nrestores all images and matrices to their original state" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 265 "" 0 "
" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Rotation" }}{EXCHG {PARA 256 ""
0 "" {TEXT -1 26 "Suppose we want the matrix" }{TEXT 256 1 " " }{TEXT
-1 110 "that rotates a point by 45 degrees around the origin (counterc
lockwise).\nWe can produce this matrix as follows" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "rotate(Pi/4); #Note: it is important to ca
pitalize \"Pi\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 292 "The matrix is
written twice. The first indicates the transformation just performed
. The second is the matrix that represents the composition of all tra
nsformations performed so far. Since only one has been performed, it \+
is the same as the first. Now let's apply this transformation again
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rotate(Pi/4);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Now the second matrix is the matr
ix of the 90 degree rotation. Notice that there is some computation e
rror. Computers are not perfect. Now let's rotate six more times:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i from 1 to 6 \n do ro
tate(Pi/4) od ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "and we see th
at we are back in starting position. The second matrix is the identit
y matrix, indicating that the composition of all the transformations a
pplied so far is the identity transformation. At any time, if you wis
h to reset back to this position so that you can start over, just type
\"" }{TEXT 257 8 "reset();" }{TEXT -1 2 "\":" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Proje
ction" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 69 "The matrix trasformatio
n of a projection is obtained by the formula \n" }{XPPEDIT 18 0 "u*u^t
/(u^t*u);" "6#*(%\"uG\"\"\")F$%\"tGF%*&)F$F'F%F$F%!\"\"" }{TEXT -1 69
" where u is the vector to be projected upon and t denotes transpose. \+
" }}{PARA 266 "" 0 "" {TEXT -1 26 "Suppose we want the matrix" }{TEXT
256 1 " " }{TEXT 270 5 "which" }{TEXT 271 1 " " }{TEXT -1 73 "projects
a point onto the line y=x.\nWe can produce this matrix as follows" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "project(1,1); " }{TEXT -1
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Now let's apply another pr
ojection, say onto the line y=2x." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "project(1,2);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Reflection" }}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 194 "This operation is similar to the project
ion transformation in that it uses the projection matrix to compute th
e reflection. Indeed, once the projection matrix is known, the reflect
ion matrix is " }{TEXT 272 4 "2P-I" }{TEXT -1 7 " where " }{TEXT 273
1 "P" }{TEXT -1 35 " denotes the projection matric and " }{TEXT 274 1
"I" }{TEXT -1 225 " denotes the identity matrix.\nThis procedure is us
ed in the following way. Say I wanted to reflect the original image ac
ross the line whose angle with the positive x-axis is 45 degrees. Firs
t we will need to reset the images." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 8 "reset();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Then \+
enter in the appropriate command," }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 33 "reflect((sqrt(2))/2,(sqrt(2))/2);" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 227 "Notice that I could have used the vector
(1,1), instead I have used one on the unit circle. It is important to
understand that the vector (sqrt(2)/2,sqrt(2)/2) and (1,1) differ onl
y in length, they both have the same direction." }}}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Matr
ix action" }}{EXCHG {PARA 267 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }
{TEXT 257 127 "These procedures allow the user to generate their own l
inear transformation simply by entering the coordinates of a 2X2 matri
x." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 267 17 "mat_row(a,
b,c,d);" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "This procedure allows \+
the user to enter in coordinates one row at a time. To generate an upp
er triangular matrix simply type" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "mat_row(1,1,0,1);" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT
-1 0 "" }{TEXT 268 20 "mat_column(a,b,c,d);" }}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 169 "This procedure is the same as the previous except that t
he user enters the coordinates one column at a time. To generate the s
ame matrix as above note it would be typed," }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 20 "mat_column(1,0,1,1);" }}}}}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 268 "" 0 "" {TEXT -1 9 "Exercise
s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 9 "Reminder:" }
{TEXT -1 59 "reset(); will restore everything to their original state.
\n\n" }{TEXT 276 2 "1)" }{TEXT -1 97 " Let T be rotation by 90 degrees
counterclockwise and let S be reflection across the line x=y.\n " }
{TEXT 277 2 "a." }{TEXT -1 68 " Compute the compositions ST and TS. Ar
e they the same?\n " }{TEXT 278 2 "b." }{TEXT -1 193 " In g
eneral the composition of a rotation with a reflection is alway
s another reflection. ST is the same as reflecion across which line? T
S is the same as reflection across which line?\n\n" }{TEXT 279 2 "2)"
}{TEXT -1 46 " Which rotations T can you think of such that " }
{XPPEDIT 18 0 "T*T*T*T*T*T;" "6#*.%\"TG\"\"\"F$F%F$F%F$F%F$F%F$F%" }
{TEXT -1 1 "=" }{XPPEDIT 18 0 "Iota;" "6#%%IotaG" }{TEXT -1 2 "?\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 2 "3)" }{TEXT -1 94 " What tr
ansformation T can you think of (other than T=I or rotation by 180 deg
rees) such that " }{XPPEDIT 18 0 "T*T;" "6#*&%\"TG\"\"\"F$F%" }{TEXT
-1 1 "=" }{XPPEDIT 18 0 "Iota;" "6#%%IotaG" }{TEXT -1 3 "?\n\n" }
{TEXT 281 2 "4)" }{TEXT -1 36 " The transformation whose matrix is " }
{XPPEDIT 18 0 "matrix([[1, a], [0, 1]]);" "6#-%'matrixG6#7$7$\"\"\"%\"
aG7$\"\"!\"\"\"" }{TEXT -1 100 " is called a shear. The larger a is, t
he greater the shear. Try it out and see why it has this name." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT
282 2 "5)" }{TEXT -1 49 " What is the transformation inverse to the sh
ear " }{XPPEDIT 18 0 "matrix([[1, 2], [0, 1]]);" "6#-%'matrixG6#7$7$\"
\"\"\"\"#7$\"\"!\"\"\"" }{TEXT -1 84 "? Try to figure it out by hand a
nd then verify by comparing the two transformations." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 2 "6)" }
{TEXT -1 106 " Let T be reflection across the line that makes a 120 de
gree angle with the positive x-axis (the line y=-(" }{XPPEDIT 18 0 "sq
rt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 67 ")x). Try to express T as a c
omposition of these transformations:\n " }{TEXT 284 2 "i)" }{TEXT -1
35 " rotation clockwise 120 degrees.\n " }{TEXT 285 0 "" }{TEXT 286
0 "" }{TEXT 287 3 "ii)" }{TEXT -1 33 " reflection across the x-axis.\n
" }{TEXT 288 4 "iii)" }{TEXT -1 32 " rotation clockwise 120 degrees.
" }}{PARA 0 "" 0 "" {TEXT -1 30 "Can a different angle be used?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT
289 2 "7)" }{TEXT -1 263 " Try to guess the matrix that reflects acros
s the line x+y=0. (The first column will be the image of (1,0) and the
second column will be the image of (0,1). Knowing this should allow y
ou to guess the answer. Then verify by letting maple compute the matri
x. Type \"" }{TEXT 290 16 "reflect(-1,1);\" " }{TEXT -1 18 "to see the
answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 291 2 "8)" }{TEXT -1
44 " A transformation whose matrix has the form " }{XPPEDIT 18 0 "matr
ix([[a, 0], [0, a]]);" "6#-%'matrixG6#7$7$%\"aG\"\"!7$F)F(" }{TEXT -1
13 " is called a " }{TEXT 292 7 "scaling" }{TEXT -1 85 " by a factor o
f a. Try it out for different values of a and see why it has this name
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 293 2 "9)" }{TEXT -1 32 " What does a matrix of the form " }
{XPPEDIT 18 0 "matrix([[a, 0], [0, b]]);" "6#-%'matrixG6#7$7$%\"aG\"\"
!7$F)%\"bG" }{TEXT -1 6 " do?\n\n" }{TEXT 294 3 "10)" }{TEXT -1 405 " \+
Notice that if the image is projected onto either the x-axis or y-axis
it seems to vanish. To see that this is not the case project the imag
e onto the y-axis and then rotate it by 45 degrees. Why does it appear
that the image has vanished when it has been transformed by a project
ion onto the line (1,1) and then projected again onto the line (-1,1) \+
(notice rotating afterward still provides no picture)?" }}}}{EXCHG
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