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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 25 "TP 4 : Espaces euclidiens" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart;with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 
"Warning, the protected names norm and trace have been redefined and u
nprotected\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "1 Orthogonalisat
ion" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 40 "Dans R^3, avec la fonction
 \"cl\351 en main\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "v1:=v
ector([1,2,-3]):v2:=vector([-1,2,-1]):v3:=vector([1,2,3]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "GramSchmidt([v1,v2,v3]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7%7%\"\"\"\"\"#!\"$7%#!#5\"\"(#\"\")F+#F&F+7
%F&F&F&" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Orthogonalisation g
\351n\351rale" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "orthogonal
isation:=proc(e,scal)\nlocal f,i;\nf:=e:\nfor i from 2 to nops(e) do f
[i]:=f[i]-add(scal(f[i],f[k])/scal(f[k],f[k])*f[k],k=1..i-1) od:\nRETU
RN(map(evalm,f))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "o
rthogonalisation([v1,v2,v3],dotprod);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7%-%'vectorG6#7%\"\"\"\"\"#!\"$-F%6#7%#!#5\"\"(#\"\")F0#F)F0-F%6#7%
F)F)F)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Dans Rn[X]" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "scal1:=(P,Q)->int(P*Q,X=-1..1):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "On a privil\351gi\351 le point d
e vue \"expression\" par rapport au point de vue \"fonction\". Attenti
on, les polynomes en jeu doivent absolument avoir X comme ind\351termi
n\351e, et pas t ou x..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"scal1(X,X^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "base_can:=[seq(X^k,k=0..6)];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)base_canG7)\"\"\"%\"XG*$)F'\"\"
#F&*$)F'\"\"$F&*$)F'\"\"%F&*$)F'\"\"&F&*$)F'\"\"'F&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "base_orth:=orthogonalisation(base_can,sca
l1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*base_orthG7)\"\"\"%\"XG,&*$
)F'\"\"#F&F&#F&\"\"$!\"\",&*$)F'F-F&F&*&#F-\"\"&F&F'F&F.,(*$)F'\"\"%F&
F&#F-\"#NF&*&#\"\"'\"\"(F&F)F&F.,(*$)F'F4F&F&*&#F4\"#@F&F'F&F&*&#\"#5
\"\"*F&F0F&F.,**$)F'F=F&F&#F4\"$J#F.*&#F4\"#6F&F*F&F&*&#\"#:FPF&F6F&F.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Vous reconnaissez les 4 premi
ers ?" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Lien avec les polynome
s de Legendre" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthop
oly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG%\"TG%
\"UG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "P(n, x) generates the nth
 Legendre polynomial. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s
eq(P(n,X),n=0..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"\"%\"XG,&*$)
F$\"\"#F##\"\"$F(#F#F(!\"\",&*$)F$F*F##\"\"&F(*&#F*F(F#F$F#F,,(*$)F$\"
\"%F##\"#N\"\")*&#\"#:F7F#F&F#F,#F*F:F#,(*$)F$F1F##\"#jF:*&#F9F7F#F.F#
F,*&#F=F:F#F$F#F#,**$)F$\"\"'F##\"$J#\"#;*&#\"$:$FNF#F5F#F,*&#\"$0\"FN
F#F'F#F##F1FNF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(rem(
base_orth[k],P(k-1,X),X),k=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)
\"\"!F#F#F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
35 3 "rem" }{TEXT -1 35 " function returns the remainder of " }{TEXT 
35 1 "a" }{TEXT -1 12 " divided by " }{TEXT 35 1 "b" }{TEXT -1 6 ". Th
e " }{TEXT 35 3 "quo" }{TEXT -1 34 " function returns the quotient of \+
" }{TEXT 35 1 "a" }{TEXT -1 12 " divided by " }{TEXT 35 1 "b" }{TEXT 
-1 16 ". The remainder " }{TEXT 35 1 "r" }{TEXT -1 14 " and quotient \+
" }{TEXT 35 1 "q" }{TEXT -1 10 " satisfy: " }{TEXT 35 11 "a = b*q + r
" }{TEXT -1 7 " where " }{TEXT 35 25 "degree(r,x) < degree(b,x)" }
{TEXT -1 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;w
ith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected na
mes norm and trace have been redefined and unprotected\n" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 55 "2 Identification de r\351flexions/project
ions orthogonales" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "A:=mat
rix([[1/3, 2/3, -2/3], [2/3, 1/3, 2/3], [-2/3, 2/3, 1/3]]):B:=matrix([
[13/14, -1/7, -3/14], [-1/7, 5/7, -3/7], [-3/14, -3/7, 5/14]]):" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Un premier exemple" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(A^2),evalm(transpose(A)-A);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F
)7%F)F)F(-F$6#7%7%F)F)F)F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 74 "kernel(A-Matrix(3,3,shape=identity)),kernel(A+Matrix(3,3,shape=i
dentity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$-%'vectorG6#7%!\"\"\"
\"!\"\"\"-F%6#7%F*F*F)<#-F%6#7%F*F(F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 172 "Autre possibilit\351, quand on a une r\351flexion, l'axe
 est l'orthogonal du plan. On peut donc trouver une base de l'axe par \+
produit vectoriel des vecteurs d'une base du plan  :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 70 "plan:=kernel(A-Matrix(3,3,shape=identity)
):crossprod(plan[1],plan[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ve
ctorG6#7%\"\"\"!\"\"F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Syst
\351matisons" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalb(transp
ose(A)=A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "Arf !" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "nops(kernel(A-transpose(A)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "L\351ger..
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "analyse_reflexion:=A-
>\nif nops(kernel(transpose(A)-A))<3\nor nops(kernel(A^2-Matrix(3,3,sh
ape=identity)))<3\nor nops(kernel(A-Matrix(3,3,shape=identity)))<>2\nt
hen false\nelse op(kernel(A+Matrix(3,3,shape=identity))) fi:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "analyse_reflexion(A);analyse
_reflexion(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"\"
!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 32 "Cas des projections orthogonales" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "analyse_projection:=A->\nif
 nops(kernel(transpose(A)-A))<3 or nops(kernel(A^2-A))<3 then false\ne
lse kernel(A-Matrix(3,3,shape=identity)) fi:" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 8 "Exemples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
245 "C:=matrix([[25/146, 33/146, -22/73], [33/146, 137/146, 6/73], [-2
2/73, 6/73, 65/73]]):Dd:=matrix([[12/13, -3/13, 4/13], [-3/13, 4/13, 1
2/13], [4/13, 12/13, -3/13]]):E:=matrix([[-10/19, 6/19, 15/19], [6/19,
 -15/19, 10/19], [15/19, 10/19, 6/19]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 131 "analyse_reflexion(C),analyse_projection(C);analyse_r
eflexion(Dd),analyse_projection(Dd);analyse_reflexion(E),analyse_proje
ction(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&falseG<$-%'vectorG6#7%
\"\"!#\"\"%\"\"$\"\"\"-F&6#7%F-#\"#6F,F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$-%'vectorG6#7%#\"\"\"\"\"$F(#!\"%F)%&falseG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%&falseGF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "evalm(E-transpose(E)),evalm(E^2-Matrix(3,3,shape=identity));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7%7%\"\"!F(F(F'F'F#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "kernel(E-Matrix(3,3,shape=id
entity)),kernel(E+Matrix(3,3,shape=identity));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$<#-%'vectorG6#7%#\"\"$\"\"#\"\"\"#\"\"&F*<$-F%6#7%\"\"!
#!\"&F*F+-F%6#7%F+#!\"$F*F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "E
 est donc la matrice d'une sym\351trie orthogonale par rapport \340 un
e droite (on parle de retournement)" }}}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart;with(linalg):" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 58 "3 Cr\351ation de matrices de r\351flexion/projection orth
ogonale" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Un premier exemple" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "v0:=vector([1,-1,1]):v:=vect
or([x,y,z]):evalm(v-2*dotprod(v,v0)/3*v0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,(%\"xG#\"\"\"\"\"$*&#\"\"#F+F*%\"yGF*F**
&#F.F+F*%\"zGF*!\"\",(F/F)*&F-F*F(F*F**&F-F*F2F*F*,(F2F)*&#F.F+F*F(F*F
3*&F-F*F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "genmatrix(
convert(%,list),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix
G6#7%7%#\"\"\"\"\"$#\"\"#F*#!\"#F*7%F+F(F+7%F-F+F(" }}}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 13 "Syst\351matisons" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 154 "generateur_reflexion:=proc(v0)\nlocal v,x,y,z;\nv:=v
ector([x,y,z]):\nRETURN(genmatrix(convert(evalm(v-2*dotprod(v,v0)/dotp
rod(v0,v0)*v0),list),[x,y,z]))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "generateur_reflexion(vector([1,-1,1]));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"\"\"\"$#\"\"#F*#!\"#F*7%F+F
(F+7%F-F+F(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Exemples" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "generateur_reflexion(vector(
[1,2,3])),generateur_reflexion(vector([1,0,1]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'matrixG6#7%7%#\"\"'\"\"(#!\"#F*#!\"$F*7%F+#\"\"$F*#!
\"'F*7%F-F2F+-F$6#7%7%\"\"!F9!\"\"7%F9\"\"\"F97%F:F9F9" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 11 "Projections" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 153 "generateur_projection:=proc(v0)\nlocal v,x,y,z;\nv:=
vector([x,y,z]):\nRETURN(genmatrix(convert(evalm(v-dotprod(v,v0)/dotpr
od(v0,v0)*v0),list),[x,y,z]))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "generateur_projection(vector([1,-1,1]));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"#\"\"$#\"\"\"F*#!\"\"F*7%F
+F(F+7%F-F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "generateur
_projection(vector([1,2,3])),generateur_projection(vector([1,0,1]));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7%7%#\"#8\"#9#!\"\"\"\"(
#!\"$F*7%F+#\"\"&F-#F/F-7%F.F3#F2F*-F$6#7%7%#\"\"\"\"\"#\"\"!#F,F<7%F=
F;F=7%F>F=F:" }}}}}}{MARK "2 3 2 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 1 1 1 33 1 1 }