Espaces euclidiens

> restart;

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

Exercice 1

> int((cos(t)-(a*t+b))^2,t=0..Pi);

Une première version brutale... Maple utilise une méthode que vous comprendrez un peu plus tard (fin d'année)

> minimize(%);

Reprenons cela dans l'esprit des espaces euclidiens...

> scal:=(f,g)->int(f*g,t=0..Pi):

Attention, f et g ne sont pas des fonctions mais des expressions en t

> solve({scal(cos(t)-a0*t-b0,1)=0,scal(cos(t)-a0*t-b0,t)=0});

> a0;

> assign(%%);

> a0;

> int((cos(t)-(a0*t+b0))^2,t=0..Pi);

> expand(%);

OK

> ortho_norm:=proc(e,scal)
local n,f,i,k:
n:=nops(e):f:=e:
for k to n do
for i to k-1 do
f[k]:=evalm(f[k]-scal(e[k],f[i])*f[i]);
od; f[k]:=evalm(f[k]/sqrt(scal(f[k],f[k]))) od:
RETURN(f) end:

Exercice 2

> f1:=vector([1,1,1]):f2:=vector([1,-1,1]):f3:=vector([1,0,0]):

> with(linalg):

> dotprod(f1,f2),dotprod(f1,f3),dotprod(f2,f3);

> n:=ortho_norm([f1,f2,f3],dotprod);

> matrix([seq([seq(dotprod(n[i],n[j]),j=1..3)],i=1..3)]);

Exercice 3

> P:=ortho_norm([1,X,X^2,X^3,X^4,X^5],(P,Q)->int(P*Q,X=-1..1));

> Q:=k->expand(diff((X^2-1)^k,X$k));

> seq(Q(i),i=1..5);

> plot({Q(1),Q(2),Q(3),Q(4)},X=-1..1);

> plot({Q(1),Q(2)},X=-1..1,-10..10);

> plot({Q(6),Q(7)},X=-1..1);

> plot({Q(6),Q(7)},X=-1..1,-50000..50000);

> matrix([seq([seq(int(Q(i)*Q(j),X=-1..1),j=1..3)],i=1..3)]);

> for i from 1 to 5 do print(quo(Q(i),P[i+1],X),rem(Q(i),P[i+1],X)) od;

> seq(sqrt(int(Q(i)*Q(i),X=-1..1)),i=1..5);

Tout va bien...

Exercice 5

> g1:=vector([1,1,1,1]):g2:=vector([1,2,3,4]):v:=vector([w,x,y,z]):

(g1,g2) est par définition une base de l'ortohogonal de F

> solve({dotprod(v-a*g1-b*g2,g1)=0,dotprod(v-a*g1-b*g2,g2)=0},{a,b});

> assign(%):

> evalm(v-2*a*g1-2*b*g2);

> ?genmatrix

The function genmatrix generates the coefficient matrix from the linear system of equations eqns in the unknowns vars. If the optional third argument ``flag'' is present, the ``right-hand side'' vector will be included as the last column of the matrix. Otherwise an optional third argument will be assigned the ``right-hand side'' vector.

> genmatrix(convert(%,list),[w,x,y,z]);

> trace(%);

Normal...

Exercice 6

> f0:=vector([1,-2,1]):v:=vector([x,y,z]):

> solve(dotprod(v-a*f0,f0)=0,{a});

> assign(%):

> evalm(v-2*a*f0);

> genmatrix(convert(%,list),[x,y,z]);

> trace(%);

OK...

Exercice 7

> f0:=vector([1,0,1,0]):v:=vector([w,x,y,z]):

f0 dirige l'orthogonal de H

> solve(dotprod(v-a*f0,f0)=0,{a});

> assign(%):

> evalm(2*a*f0-v);

> genmatrix(convert(%,list),[w,x,y,z]);

> trace(%);