︠6e610a4c-1576-4809-8b73-824e5ca0c52bs︠
typeset_mode(True)
︡a15136b8-d423-4c9a-8a84-ae414ad70d48︡︡{"done":true}
︠e4f824e3-6e1d-467c-989b-0adb0cf2f7f2i︠
%md
### Question 2
︡0733a461-c445-419c-9fbc-53dc29426f36︡︡{"done":true,"md":"### Question 2"}
︠2ea68f15-90f6-4555-86e8-a92f025a68c3s︠
A =matrix(QQ,[[2,1,-2,-1],[4,4,-3,1],[2,7,1,8]])
A.echelonize()
A
︡096b290c-acbb-4576-9c30-58fd72117ab2︡︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrr}\n1 & 0 & -\\frac{5}{4} & -\\frac{5}{4} \\\\\n0 & 1 & \\frac{1}{2} & \\frac{3}{2} \\\\\n0 & 0 & 0 & 0\n\\end{array}\\right)$
","done":false}︡{"done":true}
︠60536f28-b141-445f-a7fa-ee03ff3a8825s︠
B=matrix(QQ,[[5/4,5/4,-5],[-1/2,-3/2,10],[1,0,4],[0,1,-8]])
B.echelonize()
B
︡611d6b2c-16a9-4cbb-bfd6-c7b1d10d72b3︡︡{"html":"$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 4 \\\\\n0 & 1 & -8 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}\\right)$
","done":false}︡{"done":true}
︠baa40afd-a6ec-4af7-8412-2ed4b1fbddfai︠
%md
### Question 4
︡1e4c4798-4dfe-41ea-8a6c-4a278f6d9417︡︡{"done":true,"md":"### Question 4"}
︠68423213-8c62-41b3-a130-5e01a974f054s︠
u1=vector([3,0,4])
u2=vector([-1,0,7])
u3=vector([2,9,11])
︡4c3f8595-aee3-4ab8-a8f7-0f0f37360837︡︡{"done":true}
︠7c3883fb-7cda-4560-810c-86b4b5840d6bs︠
def proj(p,q):#la projection de p sur q
return p.dot_product(q)/q.dot_product(q) * q
︡a33c0e54-ef0b-4dc3-85ab-072d879abea6︡︡{"done":true}
︠dade5bda-ea94-480b-b864-34dfd5345100s︠
v1=u1
v2 = u2-proj(u2,v1)
v2
︡8684e44c-3bb3-45ce-8317-389e0a1aa9e4︡︡{"html":"$\\displaystyle \\left(-4,\\,0,\\,3\\right)$
","done":false}︡{"done":true}
︠8b2c190d-5fd2-439e-99db-529fda595c8fs︠
v3 = u3-proj(u3,v1)-proj(u3,v2)
v3
︡54199658-67b3-4e12-b6ed-5c9bc91fb417︡︡{"html":"$\\displaystyle \\left(0,\\,9,\\,0\\right)$
","done":false}︡{"done":true}
︠9a236b90-1c22-4d16-902e-be8bb6dbe703s︠
v2.dot_product(v3)
︡33ab39ea-45bc-4e16-aa14-be96ceea7017︡︡{"html":"$\\displaystyle 0$
","done":false}︡{"done":true}
︠be021868-7c66-4bac-8dcb-9c656be2665ci︠
%md
### Question 5
︡92af24eb-f941-4d0f-bbee-b6fa96000a0a︡︡{"done":true,"md":"### Question 5"}
︠126c0fd6-8c77-4a5a-b099-d06ec3715284s︠
f1(x) = 1
f2(x)=cos(x)
f3(x)=sin(x)
h(x)=x^2
︡cd304ae4-5ec5-477f-975c-713d47e537e8︡︡{"done":true}
︠e54c381f-33f5-4e47-8a1a-fb9ff75dd6dcs︠
g1(x) = f1(x)/sqrt(integral(f1(x)*f1(x),x,-pi,pi))
g1(x)
︡b474a4be-d144-40c5-837a-b2ab6f4cbd5a︡︡{"html":"$\\displaystyle \\frac{\\sqrt{2}}{2 \\, \\sqrt{\\pi}}$
","done":false}︡{"done":true}
︠5fc92813-c46e-4592-9334-2db3cdee7ff9s︠
g2(x) = f2(x)/sqrt(integral(f2(x)*f2(x),x,-pi,pi))
g2(x)
︡bc4fe064-628a-43b7-bb01-d9b2d9ef8790︡︡{"html":"$\\displaystyle \\frac{\\cos\\left(x\\right)}{\\sqrt{\\pi}}$
","done":false}︡{"done":true}
︠5c789add-4686-49a8-a36c-8cab15eaba58s︠
g3(x) = f3(x)/sqrt(integral(f3(x)*f3(x),x,-pi,pi))
g3(x)
︡667c2e4d-c02e-4a5e-900d-d19d38c86247︡︡{"html":"$\\displaystyle \\frac{\\sin\\left(x\\right)}{\\sqrt{\\pi}}$
","done":false}︡{"done":true}
︠02e831c8-fb3c-4b25-af70-bc5e6b2ffddes︠
︡4b9fd02e-d851-47bd-9ee2-384f877399a3︡︡{"done":true}
︠190cd173-ff60-43b9-8c31-edfccac5a9d6s︠
︡e6fba2d5-31b0-4ad8-8663-17fbe299b8f4︡︡{"done":true}
︠76779bd6-dd5b-4d21-8fac-4bb0b7127689s︠
c1 = integral(g1(x)*h(x),x,-pi,pi)
c1
c2 = integral(g2(x)*h(x),x,-pi,pi)
c2
c3 = integral(g3(x)*h(x),x,-pi,pi)
c3
︡0111ad60-9c83-43c0-996b-0a55f45b4dbb︡︡{"html":"$\\displaystyle \\frac{1}{3} \\, \\sqrt{2} \\pi^{\\frac{5}{2}}$
","done":false}︡{"html":"$\\displaystyle -4 \\, \\sqrt{\\pi}$
","done":false}︡{"html":"$\\displaystyle 0$
","done":false}︡{"done":true}
︠42114af6-7ca3-4599-a836-82a48deb6a75s︠
H(x) = c1 * g1(x) + c2*g2(x)
CourbeH = plot(H(x),x,-pi,pi)
Courbeh = plot(h(x),x,-pi,pi, color="red")
C1 = plot(f1(x),x,-pi,pi, color="darkgreen", linestyle="dotted")
C2 = plot(f2(x),x,-pi,pi, color="darkgreen", linestyle="dotted")
C3 = plot(f3(x),x,-pi,pi, color="darkgreen", linestyle="dotted")
show(CourbeH + Courbeh + C1+C2+C3, figsize = 4)
︡f0472829-04c4-4b5a-a9a8-9662ba50dc66︡︡{"once":false,"done":false,"file":{"show":true,"uuid":"404b8bc7-fcdd-41d0-acb1-7d9b87261f3a","filename":"/projects/b2ab33d7-45ee-4e82-8abc-a3432865ad77/.sage/temp/compute3-us/854/tmp_1i8RoI.svg"}}︡{"html":"","done":false}︡{"done":true}
︠33dff81e-bf33-4e8f-b904-13e4eef50bbci︠
%md
La courbe bleue représente la projection de $h(x)$ sur l'espace engendré par les trois fonctions données dans l'énoncé. Il s'agit de la courbe qui peut approcher le mieux la parabole rouge, parmi toutes les courbes que l'on peut construire en faisant des combinaisons linéaires des trois fonctions données (en pointillés verts)
︡ef823531-03cb-4881-8f84-a914fdb28dee︡︡{"done":true,"md":"La courbe bleue représente la projection de $h(x)$ sur l'espace engendré par les trois fonctions données dans l'énoncé. Il s'agit de la courbe qui peut approcher le mieux la parabole rouge, parmi toutes les courbes que l'on peut construire en faisant des combinaisons linéaires des trois fonctions données (en pointillés verts)"}
︠fb4548b6-0f0e-4bf9-8376-ad35899ffabes︠
︡11bdb568-bbb0-40e2-9fc2-21147a4f2291︡︡{"done":true}
︠2c17e78a-3e9a-4ecf-b613-e73eb5839cb4i︠
%md
### Question 6
︡388b876d-c274-4ddf-8851-e9b5fadb99b9︡︡{"done":true,"md":"### Question 6"}
︠df087a42-6bc7-429c-b148-fa759d453a42s︠
def ps(p,q):
return integrate(exp(-x^2)* p(x)*q(x),x,-infinity,infinity)
def pproj(p,q):
return ps(p,q)/ps(q,q)*q
︡88930333-8113-4312-9efa-3279fa6b7079︡︡{"done":true}
︠7e7cc0a7-6336-49b7-b989-48ad3e07ca3ds︠
u0(x) = 1
u1(x) = x
u2(x) = x^2
︡4bc698db-557f-444b-9554-d5e4c11f5eff︡︡{"done":true}
︠e28e7f7c-feeb-451a-827b-0bad8f0daab5s︠
f0(x) = u0(x)
f0(x)
︡e9aa496f-f17b-45d7-a589-6af8b6ac6efd︡︡{"html":"$\\displaystyle 1$
","done":false}︡{"done":true}
︠b07ddc8a-c256-4eab-9515-c3365467d0a0s︠
ps(f0,f0)
︡fe227a96-64fc-4040-8f5f-34fa50b12f65︡︡{"html":"$\\displaystyle \\sqrt{\\pi}$
","done":false}︡{"done":true}
︠f659beee-9862-4c48-84e8-e4180ffc240fs︠
f1(x) = u1- pproj(u1,f0)
f1(x)
︡e2c32b14-3540-42d3-a43d-104dba1ea59e︡︡{"html":"$\\displaystyle x$
","done":false}︡{"done":true}
︠ce7a1948-ad90-4ae9-8d5c-3e95da8e9b1bs︠
ps(f1,f1)
︡64382901-42eb-479b-926d-12f16e0fda43︡︡{"html":"$\\displaystyle \\frac{1}{2} \\, \\sqrt{\\pi}$
","done":false}︡{"done":true}
︠a74154b7-e6e8-4d8d-af24-b6d84b4f216as︠
f2(x) = u2 - pproj(u2,f0) - pproj(u2,f1)
f2(x)
︡1c88f2ef-3f9c-419a-bdb5-28d39a95ce75︡︡{"html":"$\\displaystyle x^{2} - \\frac{1}{2}$
","done":false}︡{"done":true}
︠dd90ca04-3f84-497b-a5e4-7d58cc7259cfs︠
ps(f2,f2)
︡f076cc66-17e5-4b61-9d9a-5331c8f64efc︡︡{"html":"$\\displaystyle \\frac{1}{2} \\, \\sqrt{\\pi}$
","done":false}︡{"done":true}
︠772693e3-2b6c-40b8-9ec9-2ec15ffa0823︠
︠cd9f0bef-cbb3-4a5e-b61e-12b4ef292ac6︠