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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT -1 44 "Introduction to \+
Eigenvalues and Eigenvectors" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }
{TEXT 301 0 "" }{TEXT 302 22 "Linear Algebra Project" }}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 10 "Objective:" }}{PARA 0 "" 0 "" 
{TEXT -1 152 "To understand the concepts of eigenvalue and eigenvector
; to be able to find the eigenvalues and eigenvectors (if any) of a gi
ven linear transformation." }}{PARA 0 "" 0 "" {TEXT -1 150 "NOTE: Befo
re doing this worksheet, it is important  first to work through and un
derstand the labs on linear transformations of the plane and of space.
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 276 23 "Background I
nformation:" }}{PARA 0 "" 0 "" {TEXT -1 64 "Recall that if A is the ma
trix of a linear transformation, and A" }{TEXT 277 2 "v " }{TEXT -1 2 
"= " }{TEXT 278 1 "w" }{TEXT -1 25 ", we say that the vector " }{TEXT 
279 1 "v" }{TEXT -1 10 " has been " }{TEXT 280 11 "transformed" }
{TEXT -1 17 " into the vector " }{TEXT 281 1 "w" }{TEXT -1 28 ". It so
metimes happens that " }{TEXT 282 1 "w" }{TEXT -1 6 " is a " }{TEXT 
283 15 "scalar multiple" }{TEXT -1 4 " of " }{TEXT 284 1 "v" }{TEXT 
-1 18 ", in other words, " }{TEXT 285 1 "w" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 286 1 "v
" }{TEXT -1 31 " for some specific real number " }{XPPEDIT 18 0 "lambd
a;" "6#%'lambdaG" }{TEXT -1 28 ". In that case, we say that " }{TEXT 
287 1 "v" }{TEXT -1 7 " is an " }{TEXT 288 11 "eigenvector" }{TEXT -1 
33 " of the transformation, and that " }{XPPEDIT 18 0 "lambda;" "6#%'l
ambdaG" }{TEXT -1 7 " is an " }{TEXT 289 10 "eigenvalue" }{TEXT -1 24 
" of the transformation. " }{TEXT 294 6 "NOTE: " }{TEXT -1 34 "This de
finition applies only when " }{TEXT 295 1 "v" }{TEXT -1 6 " is a " }
{TEXT 296 8 "non-zero" }{TEXT -1 13 " vector. (If " }{TEXT 297 2 "v " 
}{TEXT -1 35 "is the zero vector then of course A" }{TEXT 298 1 "0" }
{TEXT -1 3 " = " }{TEXT 299 1 "0" }{TEXT -1 10 " and thus " }{XPPEDIT 
18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 150 " could have any value. The
 eigenvalue would not be a unique number.) It is possible, however for
 the number 0 to be an eigenvalue of a transformation." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Some examples: 1) Co
nsider the dilation transformation where  " }{TEXT 300 2 "r " }{TEXT 
-1 30 "= 5. For this transformation, " }{TEXT 290 3 "all" }{TEXT -1 
170 " non-zero vectors are eigenvectors with eigenvalue 5.  2) Conside
r the transformation that reflects every vector across the line y = x.
 In this transformation, there are " }{TEXT 291 10 "two groups" }
{TEXT -1 187 " of eigenvectors: i) vectors lying on the line  y = x ar
e eigenvectors with eigenvalue 1;  ii) vectors lying on the line y = -
x are eigenvectors with eigenvalue -1. Any vector that is in " }{TEXT 
293 7 "neither" }{TEXT -1 20 " of these groups is " }{TEXT 292 3 "not
" }{TEXT -1 19 " an eigenvector.   " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }{TEXT 257 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 123 "a) Use geometric reasoning to find a
ll eigenvectors, and their corresponding eigenvalues, of the following
 transformations:" }}{PARA 0 "" 0 "" {TEXT -1 123 "    i)  The transfo
rmation that rotates all vectors counterclockwise 60 degrees around th
e origin.                         " }}{PARA 0 "" 0 "" {TEXT -1 118 "  \+
  ii) The transformation that projects every vector onto the line maki
ng a 60 degree angle with the positive x-axis." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "b) Use the LAMP command \+
\"Clock\" and/or \"Headtail\"  in step-by-step mode to verify your ans
wers to part a)." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
0 "" }{TEXT 259 9 "Solution:" }{TEXT 260 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "b) We wil
l first activate LAMP and use the commands to generate the matrices of
 the required transformations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "with (linalg): with (plots): with (Lamp);\n" }}{PARA 7 "" 1 "
" {TEXT -1 80 "Warning, the protected names norm and trace have been r
edefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, th
e name changecoords has been redefined\n" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7_p%'Axis3dG%*BackidmatG%*BacksolveG%(BandmatG%*BasisgridG%$BugG
%&Bug3dG%'Bug3dhG%)CharpolyG%+Chevron3dhG%&ClockG%*ColorlistG%.Compone
ntplotG%)CopyintoG%*CrossprodG%'Cube3dG%(Cube3dhG%$DetG%(DiagmatG%(Dot
prodG%*DrawlinesG%+DrawmatrixG%+DrawplanesG%(DrawvecG%*Drawvec3dG%&Eva
lmG%(EvaluesG%)EvectorsG%'ExpandG%*GenmatrixG%)GetcolorG%%GridG%)Gridg
ameG%*GridvectsG%)HeadtailG%(Hotel3dG%)Hotel3dhG%&HouseG%)House3dhG%&I
dmatG%(InverseG%&Jet3dG%'Jet3dhG%*JordanmatG%(Lamp3dhG%&LamphG%(Letter
LG%(LetterNG%(LetterSG%(LetterXG%$MagG%)MapcolorG%,MappicturesG%/Matri
xtovectorG%)MatsolveG%&MovieG%*NullbasisG%%PathG%(ProjectG%+Projectmat
G%-Projectmat3dG%)QuestionG%(RandmatG%'ReduceG%+ReflectmatG%-Reflectma
t3dG%*ResidualsG%*RotatematG%,Rotatemat3dG%&RowopG%+SymbollistG%+Traje
ctoryG%-Trajectory3dG%*TransformG%-TranslatematG%/Translatemat3dG%)Uni
tspanG%*VandermatG%+VectorgridG%+VectorlineG%-VectranslateG%*Xshearmat
G%*YshearmatG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "R := Rotat
emat(Pi/3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'RTABLEG6%\")
Gf$)>-%'MATRIXG6#7$7$#\"\"\"\"\"#,$*&F0!\"\"\"\"$F.F37$,$*&F0F3F4F.F/F
.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P := Projectm
at(Pi/3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'RTABLEG6%\")wV
Q>-%'MATRIXG6#7$7$#\"\"\"\"\"%,$*&F0!\"\"\"\"$#F/\"\"#F/7$F1#F4F0%'Mat
rixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Clock(R);\n" }}
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" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "C
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 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40
" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "C
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" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "C
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" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "C
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Notice where
 the red and blue vectors are in the same direction, and also where th
e blue vector disappears!" }}}}{PARA 0 "" 0 "" {TEXT -1 77 "__________
___________________________________________________________________" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }
{TEXT 261 0 "" }{TEXT 262 10 "ASSIGNMENT" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 263 10 "Problem 1:" }{TEXT 267 0 "" }{TEXT 268 0 
"" }{TEXT 269 0 "" }{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }
{TEXT 273 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "a) Use geometric reason
ing to find all eigenvectors, and their corresponding eigenvalues, of \+
the following transformations:" }}{PARA 0 "" 0 "" {TEXT -1 124 "    i)
  The transformation that rotates all vectors counterclockwise 180 deg
rees around the origin.                         " }}{PARA 0 "" 0 "" 
{TEXT -1 120 "    ii) The transformation that reflects every vector ac
ross the line making a 60 degree angle with the positive x-axis." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "b) Use t
he LAMP command \"Clock\" and/or \"Headtail\"  in step-by-step mode to
 verify your answers to part a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 264 0 "" }{TEXT 265 0 "" }{TEXT 266 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 0 "" }{TEXT 274 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT 
-1 268 "In three dimensions, what are the eigenvectors and their corre
sponding eigenvalues for the transformation that reflects all vectors \+
across the yz-plane? (Use geometric reasoning.)                       \+
                                                                  " }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 275 10 "Problem 3:" }}
{PARA 0 "" 0 "" {TEXT -1 174 "In three dimensions, what are the eigenv
ectors and their corresponding eigenvalues for the transformation that
 rotates vectors 180 degrees counterclockwise around the z-axis?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 74 "________
__________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 94 "MSIP Grant #P120A80089-98: \"Three Urban \+
Calculus Reform Programs: Adopting the Best\" 1998-2001" }}}{MARK "1 2
" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }
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