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1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[r%$AddG%(AdjointG%3BackwardSubst
ituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-BilinearF
ormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0Colum
nDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0Conditi
onNumberG%/ConstantMatrixG%/ConstantVectorG%2CreatePermutationG%-Cross
ProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/DiagonalMatrixG%*Di
mensionG%+DimensionsG%+DotProductG%6EigenConditionNumbersG%,Eigenvalue
sG%-EigenvectorsG%&EqualG%2ForwardSubstituteG%.FrobeniusFormG%4Gaussia
nEliminationG%2GenerateEquationsG%/GenerateMatrixG%2GetResultDataTypeG
%/GetResultShapeG%5GivensRotationMatrixG%,GramSchmidtG%-HankelMatrixG%
,HermiteFormG%3HermitianTransposeG%/HessenbergFormG%.HilbertMatrixG%2H
ouseholderMatrixG%/IdentityMatrixG%2IntersectionBasisG%+IsDefiniteG%-I
sOrthogonalG%*IsSimilarG%*IsUnitaryG%2JordanBlockMatrixG%+JordanFormG%
(LA_MainG%0LUDecompositionG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%
*MatrixAddG%.MatrixInverseG%5MatrixMatrixMultiplyG%+MatrixNormG%5Matri
xScalarMultiplyG%5MatrixVectorMultiplyG%2MinimalPolynomialG%&MinorG%(M
odularG%)MultiplyG%,NoUserValueG%%NormG%*NormalizeG%*NullSpaceG%3Outer
ProductMatrixG%*PermanentG%&PivotG%*PopovFormG%0QRDecompositionG%-Rand
omMatrixG%-RandomVectorG%%RankG%6ReducedRowEchelonFormG%$RowG%-RowDime
nsionG%-RowOperationG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-Scal
arVectorG%*SchurFormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVec
torG%)SumBasisG%0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG
%0TridiagonalFormG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,Vecto
rAngleG%5VectorMatrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+Ze
roMatrixG%+ZeroVectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c
hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
246 "Consider the vectors a1 = <1, 0, 1>, a2 = <0, 1, 0>, and a4 = <1,
1, -1>. Are these vectors orthogonal? Since they are three-dimensional
, they should be at right angles to each other. Can you decide based o
n the following 3D plot? Rotate the plot." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 196 "a1 := arrow(<1,0,1>, shape=arrow, color=red):\na2 \+
:= arrow(<0,1,0>, shape=arrow, color=blue):\na4 := arrow(<1,1,-1>, sha
pe=arrow, color=green):\ndisplay(a1, a2, a4, scaling=CONSTRAINED, axes
=FRAMED);" }}{PARA 13 "" 1 "" {GLPLOT3D 324 324 324 {PLOTDATA 3 "6'-%'
CURVESG6&7$7%$\"\"!F)F(F(7%$\"+++++5!\"*F(F+7%7%$\"+.+++v!#5F($\"+.+++
&)F2F*7%F3F(F0-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\"
)F(F(-F$6&7$F'7%F(F+F(7%7%F($\"+++++!)F2$!+++++]!#6FD7%F(FG$\"+++++]FK
F6-F;6&F=F(F(F>-F$6&7$7%F(F($!\"!F)7%$\"+**********F2FX$!+**********F2
7%7%$\"+1mWYwF2Fhn$!+yn52()F2FW7%$\"+)Q`NN)F2F]o$!+;K*GH(F2F6-F;6&F=F(
F>F(-%*AXESSTYLEG6#%&FRAMEG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 
2 1 1 3 1 1.000000 60.000000 6.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "Consider the vectors b1 = <
3, 1, 1>, b2 = <-1, 2, 1>, and b4 = <-1/2,-2, 7/2>. Are these vectors \+
orthogonal? Since they are three-dimensional, they should be at right \+
angles to each other. Can you decide based on the following 3D plot? R
otate the plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "b1 := a
rrow(<3,1,1>, shape=arrow, color=red):\nb2 := arrow(<-1,2,1>, shape=ar
row, color=blue):\nb4 := arrow(<-0.5,-2,7/2>, shape=arrow, color=green
):\ndisplay(b1, b2, b4, scaling=CONSTRAINED, axes=FRAMED);" }}{PARA 
13 "" 1 "" {GLPLOT3D 324 324 324 {PLOTDATA 3 "6'-%'CURVESG6&7$7%$\"\"!
F)F(F(7%$\"+++++I!\"*$\"+++++5F-F.7%7%$\"+Nec_BF-$\"+<h)=%y!#5$\"+I)Q6
e*F6F*7%$\"+lTVZCF-$\"+$)Q6e\")F6$\"+q6')=kF6-%&STYLEG6#%,PATCHNOGRIDG
-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7%$!+**********F6$\"+++++
?F-$\"+**********F67%7%$!++KRwxF6$\"+S'y_b\"F-$\"+()R.=\"*F6FN7%$!+'z1
OA)F6$\"+g8sW;F-$\"+4g'>)oF6F@-FE6&FGF(F(FH-F$6&7$F'7%$!+++++]F6$!++++
+?F-$\"+++++NF-7%7%$!+dEcvNF6$!+i]AI9F-$\"+Tw2.HF-Fco7%$!+XtVCWF6$!+Q
\\xp<F-$\"+fB#pp#F-F@-FE6&FGF(FHF(-%*AXESSTYLEG6#%&FRAMEG-%(SCALINGG6#
%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 3 1 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "Next verify they are othogonal." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "b1 := Vector[column]([3,1,1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b1G-%'RTABLEG6%\");uv6-%'MATRIXG6#7%7#\"\"$7#\"\"\"F
/&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "b
2 := Vector[column]([-1,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b
2G-%'RTABLEG6%\")'*po6-%'MATRIXG6#7%7#!\"\"7#\"\"#7#\"\"\"&%'VectorG6#
%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "b3 := Vector[c
olumn]([-0.5,-2,7/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G-%'RTA
BLEG6%\")GkP9-%'MATRIXG6#7%7#$!\"&!\"\"7#!\"#7##\"\"(\"\"#&%'VectorG6#
%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "b1.b2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "b1.b3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "b2.b3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "
w1 := <2,1,0>:\nw2 := <1,0,2>:\nw3 := <0,-2,1>:\nord := GramSchmidt([w
1,w2,w3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ordG7%-%'RTABLEG6%\")
/#*o9-%'MATRIXG6#7%7#\"\"#7#\"\"\"7#\"\"!&%'VectorG6#%'columnG-F'6%\")
7D(G\"-F+6#7%7##F1\"\"&7##!\"#F@F.F4-F'6%\")Wh%H\"-F+6#7%7##F/\"\"$7##
!\"%FL7##!\"\"FLF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DotPr
oduct(ord[2],ord[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DotProduct(ord[1],ord[3]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 192 "c1 := arrow(ord[1], shape=arrow, color=red):\nc2 := \+
arrow(ord[2], shape=arrow, color=blue):\nc4 := arrow(ord[3], shape=arr
ow, color=green):\ndisplay(c1, c2, c4, scaling=CONSTRAINED, axes=FRAME
D);" }}{PARA 13 "" 1 "" {GLPLOT3D 321 321 321 {PLOTDATA 3 "6'-%'CURVES
G6&7$7%$\"\"!F)F(F(7%$\"+++++?!\"*$\"+++++5F-F(7%7%$\"++++];F-$\"+-+++
q!#5F(F*7%$\"++++]:F-$\"+-+++!*F6F(-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG
6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7%$F,F6$!+++++SF6F+7%7%$\"+/ky_6F6$
!+3Gd0BF6$\"+!ogBi\"F-FJ7%$\"+'f8s/#F6$!+!>FW4%F6$\"+?$Rwd\"F-F<-FA6&F
CF(F(FD-F$6&7$7%F(F($!\"!F)7%$\"+nmmmmF6$!+LLLL8F-$!+MLLLLF67%7%$\"+Ox
ze_F6$!+Z&f<0\"F-$!+gE-7MF6F_o7%$\"+M*oyS&F6$!+(yt:3\"F-$!+w1J@>F6F<-F
A6&FCF(FDF(-%*AXESSTYLEG6#%&FRAMEG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 
1 10 0 2 1 1 3 1 1.000000 81.000000 32.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ordN:=GramSchm
idt(\{w1,w2,w3\},normalized);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%or
dNG<%-%'RTABLEG6%\")c!3Z\"-%'MATRIXG6#7%7#,$*(\"\"#\"\"\"\"$0\"!\"\"F3
#F2F1F47#,$*(F1F2\"#@F4F3F5F47#,$*&F3F4F3F5F2&%'VectorG6#%'columnG-F'6
%\")3M*\\\"-F+6#7%7#,$*(\"\"%F2F9F4F9F5F27#,$*&F9F4F9F5F47#,$*(F1F2F9F
4F9F5F4F=-F'6%\")_`'f\"-F+6#7%7#,$*&\"\"&F4FZF5F27#\"\"!7#,$*(F1F2FZF4
FZF5F2F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "d1 := arrow(or
dN[1], shape=arrow, color=red):\nd2 := arrow(ordN[2], shape=arrow, col
or=blue):\nd4 := arrow(ordN[3], shape=arrow, color=green):\ndisplay(d1
, d2, d4, scaling=CONSTRAINED, axes=FRAMED);" }}{PARA 13 "" 1 "" 
{GLPLOT3D 321 321 321 {PLOTDATA 3 "6'-%'CURVESG6&7$7%$\"\"!F)F(F(7%$!+
Z,!=&>!#5$!+M2+f(*F-$\"+M2+f(*!#67%7%$!+I1(=b\"F-$!+\\JNfxF-$\"+5MLy7F
-F*7%$!+0'45d\"F-$!+C![]&yF-$\"+qq1JGF2-%&STYLEG6#%,PATCHNOGRIDG-%'COL
OURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$7%F(F($!\"!F)7%$\"+6crG()F-$!+-*y
@=#F-$!+0yNkVF-7%7%$\"+Q+FrnF-$!+5v\"Gp\"F-$!+\\RNTRF-FS7%$\"+R\\n%>(F
-$!+N(o')z\"F-$!+Q&=;/$F-FB-FG6&FIF(F(FJ-F$6&7$F'7%$\"+af8sWF-F($\"+3>
FW*)F-7%7%$\"+o^\\IJF-F($\"+CV-ztF-Fho7%$\"+fB#\\-%F-F($\"+H2\"=$pF-FB
-FG6&FIF(FJF(-%*AXESSTYLEG6#%&FRAMEG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 
0 1 10 0 2 1 1 3 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Are the vectors d1
, d2, and d4 normalized verisons of c1, c2, and c4?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "19 0 0" 55 }{VIEWOPTS 1 1 0 
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