Chapitres Maths en ECG1
Chapitres Maths en ECG1
Exercices et Corrigés : Espaces vectoriels en ECG1
Résumé de cours Exercices Corrigés
Cours en ligne de Maths en ECG1
Corrigés – Espaces vectoriels et applications linéaires
Exercice 1 :
1) Linéarité :
Pour montrer que est linéaire, on se donne deux triplets
et un réel
Montrons que
![Rendered by QuickLaTeX.com f \left( \left( x_1 , y_1 , z_1 \right) + \lambda \left( x_2 , y_2 , z_2 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-795aeb9c78f4a7bc6161bb21116a39d4_l3.png)
![Rendered by QuickLaTeX.com = f \left( x_1 , y_1 , z_1 \right) + \lambda f \left( x_2 , y_2 , z_2 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b44f479b9b3da8bbf64b7aecd25128e2_l3.png)
Il suffit d’écrire les choses !
![Rendered by QuickLaTeX.com = ( x_1 + \lambda x_2 - \left( y_1 + \lambda y_2 \right) - \left( z_1 + \lambda z_2 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0de70f6b865d6a578f4b27196ff70056_l3.png)
![Rendered by QuickLaTeX.com , y_1 + \lambda y_2 - 2 \left( z_1 + \lambda z_2 \right) )](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e3393457b39e124515efd882287be82b_l3.png)
![Rendered by QuickLaTeX.com = ( x_1 - y_1 - z_1 + \lambda \left( x_2 - y_2 - z_2 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a1e5b7fa9dd6fb0e16d9757f092014b4_l3.png)
![Rendered by QuickLaTeX.com , y_1 - 2 z_1+ \lambda \left( y_2 - 2 z_2 \right) )](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9ebec02c7728c1420d0ec1ee95086ce2_l3.png)
![Rendered by QuickLaTeX.com = \left( x_1 - y_1 - z_1 , y_1 - 2 z_1 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-70ee8fe349602cb137d805e5b7059dbc_l3.png)
![Rendered by QuickLaTeX.com + \lambda \left( x_2 , y_2 - z_2 , y_2 - 2 z_2 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-69b7a1faa1d59fcbfa29cc992baccb3d_l3.png)
![Rendered by QuickLaTeX.com = f \left( x_1 , y_1 , z_1 \right) + \lambda f \left( x_2 , y_2 , z_2 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-637c85ec0930ce8b895f73713be5060c_l3.png)
![Rendered by QuickLaTeX.com \left( x, y , z \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-02d0d48acf730b71b2eb063993dc9d30_l3.png)
![Rendered by QuickLaTeX.com f \left( x , y , z \right) = \left( 0 , 0 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c00ec605763ec195a8209dcee70ebbea_l3.png)
![Rendered by QuickLaTeX.com \left( x , y , z \right) \in \mathrm{Ker} \left( f \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3973866e5389a14f0fbfa1d5585a5b92_l3.png)
![Rendered by QuickLaTeX.com \Leftrightarrow \begin{cases} x - y - z & = 0 \\ y - 2 z& = 0 \end{cases}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0f2f3ec1f35cfa80b9a4a2bf604ce5b5_l3.png)
![Rendered by QuickLaTeX.com \Leftrightarrow \begin{cases} y & = 2 z \\ x & = y + z \end{cases}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-06b4119903ebd9b3f20a87aa1744906c_l3.png)
![Rendered by QuickLaTeX.com \Leftrightarrow \begin{cases} y & = 2 z \\ x & = 3z \end{cases}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fda675afe36748af71cfc03e76fd38da_l3.png)
![Rendered by QuickLaTeX.com \left( x , y , z \right) = \left( 3 z, 2 z , z \right) = z \left( 3 , 2, 1 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d5e69d69e34de56589e9df5efc5a6209_l3.png)
![Rendered by QuickLaTeX.com \mathrm{Ker} \left( f \right) = \mathrm{Vect} \left( \left( 3 , 2 , 1 \right) \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-098dd2a596685d9fd454912fc7f3908b_l3.png)
![Rendered by QuickLaTeX.com 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cef179a64f3446ea0212dc931dea6fc8_l3.png)
Comme on a
Ainsi
et
donc
si, et seulement si, Il s’ensuit que
est racine d’ordre au moins
pour
Comme
il s’ensuit que
On a montré que
l’inclusion réciproque étant claire, on a bien montré que
Image : On commence par appliquer le théorème du rang :
Ainsi Finalement
et
donc
En particulier,
est bijective.
3) Linéarité :
La linéarité est laissé au lecteur.
![Rendered by QuickLaTeX.com \left( x , y \right) \in \mathbb{R}^2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-72caf23077a2e240b9f05dcb7c79e23b_l3.png)
![Rendered by QuickLaTeX.com \left( x , y \right) \in \mathrm{Ker} \left( h \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-dd7eef804384fca57d0dba90308c3f64_l3.png)
![Rendered by QuickLaTeX.com h \left( x , y \right) = \left( y , x \right) = \left( 0 , 0 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-11c35d1725995f2f403b9bb59391991b_l3.png)
![Rendered by QuickLaTeX.com x = y = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-30729e60468749b653656ec1fc867b0e_l3.png)
![Rendered by QuickLaTeX.com \mathrm{Ker} \left( h \right) = \left\{ \left( 0 , 0 \right) \right\}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5b1cf7221a2858ac695b3142a157d93d_l3.png)
![Rendered by QuickLaTeX.com h](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png)
Ainsi et donc comme ci-dessus
et
est surjective.
![Rendered by QuickLaTeX.com h](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png)
COURS DE MATHS
Les meilleurs professeurs particuliers
Pour progresser et réussir
Avis Google France ★★★★★ 4,9 sur 5
Exercice 2 :
1) Soit alors
2) Pour trouver les antécédents éventuels de
on résout l’équation
On récupère le système
La résolution de ce système se fait grâce au pivot de Gauss. On trouve
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com \left( - 2 , 1 , 3 \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-901aa1f4fa1c0339216f8c703ac0f38d_l3.png)
![Rendered by QuickLaTeX.com f \left( u \right) = \left( - 1 , - 1, 8 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f49bfb9f9af0bd51ba4155f1c2feb8a5_l3.png)
La résolution de ce système se fait grâce au pivot de Gauss. On trouve
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
![Rendered by QuickLaTeX.com \mathrm{Ker} \left( f \right) = \left\{ \left( 0 , 0 , 0 \right) \right\}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-86c923807823f86b62ac0cb231714c87_l3.png)
![Rendered by QuickLaTeX.com \left( x , y, z \right) \in \mathrm{Ker} \left( f \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6249e766a6009b08aa20d0102160eaa2_l3.png)
![Rendered by QuickLaTeX.com f \left( x , y , z \right) = \left( 0 , 0 , 0 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-22517f4c08fa69fdad11705d1fceb7b7_l3.png)
La résolution donne Ainsi
L’inclusion réciproque étant claire, on a établi que
et
est injective.
Or donc
Et,
et ces deux espaces ont la même dimension, ils sont donc égaux. Donc
est surjective.
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
Exercice 3 :
1) Soient et
On a :
est donc linéaire.
2) Soit définie par
![Rendered by QuickLaTeX.com f \circ g.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0d894ea50b836d1aa896796d19386fbd_l3.png)
![Rendered by QuickLaTeX.com M \in \mathcal M_n \left( \mathbb{R} \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c6be735407b6fd85aa50f2c19e92bbe3_l3.png)
Il s’ensuit que Donc
est bijective et
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
![Rendered by QuickLaTeX.com I_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7602cca9692cafb9ac7ba839b675750f_l3.png)
![Rendered by QuickLaTeX.com f,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-05727159a31fc466cbfe67c239a3ca36_l3.png)
![Rendered by QuickLaTeX.com N \in \mathcal M_n \left( \mathbb{R} \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-21edc2973a96880f09d908d41f15d558_l3.png)
Donc est inversible et
COURS PARTICULIERS EN LIGNE
Nous avons sélectionné pour vous les meilleurs professeurs particuliers.
POUR ACCÉLÉRER MA PROGRESSION
Avis Google France ★★★★★ 4,9 sur 5
Exercice 4 :
1) Déjà est non vide car la suite nulle est bien dans
![Rendered by QuickLaTeX.com \left( u_n \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d970b9eae22b2141696aa5504c9fe624_l3.png)
![Rendered by QuickLaTeX.com \left( v_n \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-95e255edc85e697ab2852b94ab581997_l3.png)
![Rendered by QuickLaTeX.com E](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png)
![Rendered by QuickLaTeX.com \lambda \in \mathbb{R}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e965b55dc00b09b5c1f66f2b811b3bce_l3.png)
![Rendered by QuickLaTeX.com u + \lambda v \in E.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-baad7b66cda5cabe27484cf0ab6428a4_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bcc659f188465fe4226d2f333afa1af3_l3.png)
![Rendered by QuickLaTeX.com \left( u_{n + 2 } + \lambda v_{n + 2 } \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d1d6fba1638133abe12befbffb4b8696_l3.png)
![Rendered by QuickLaTeX.com = u_{n + 2 } + \lambda v_{n + 2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a1312dd5096012d88e94100a68ea95ba_l3.png)
![Rendered by QuickLaTeX.com = 3 u_{n + 1 } - 2 u_n + \lambda \left( 3 v_{n + 1} - 2 v_n \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-57fcb3e42c2706518d2e6240d0127073_l3.png)
![Rendered by QuickLaTeX.com = 3 \left( u_{n + 1} + \lambda v_{n + 1} \right) - 2 \left( u_n + \lambda v_n \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-457ee561ba438a55bc73ce56370b3712_l3.png)
![Rendered by QuickLaTeX.com u + \lambda v \in E.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-baad7b66cda5cabe27484cf0ab6428a4_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com u \in E](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-efa03fa7c40bb7d27b46b884941aa569_l3.png)
![Rendered by QuickLaTeX.com \varphi \left( u \right) = \left( u_0 , u_1 \right) = \left( 0 , 0 \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-67ff699dbff6c368be78ba267c759853_l3.png)
![Rendered by QuickLaTeX.com u = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ebe08819538cd9e42ec4b57c6c154885_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N}, u_n = 0 .](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cfba3916816fc3ed8a0626df4d4a0090_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-05ae0ad15f88e28f167e42a79bd14ab6_l3.png)
![Rendered by QuickLaTeX.com u_n = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c74898844b8e65876191bb8a4ecbb0ca_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-67fb518d1408224ef4168d7d30b65c45_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9a76ed9903513c05317b1ba218a4e451_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-05ae0ad15f88e28f167e42a79bd14ab6_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_{n + 1}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b3bcbc7664db9a0cbd1e3dcd233c2d41_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_{n + 2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d7c4d78d4a74c3b4359768b595b68f7e_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-05ae0ad15f88e28f167e42a79bd14ab6_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_{n + 1}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b3bcbc7664db9a0cbd1e3dcd233c2d41_l3.png)
![Rendered by QuickLaTeX.com u_n = u_{n + 1} = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fdafa382cf80b70b495174e5b23660a1_l3.png)
![Rendered by QuickLaTeX.com u_{n + 2 } = 3 u_{n + 1 } - 2 u_n = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bcf0f38c625b9b831628cace2c552ce0_l3.png)
![Rendered by QuickLaTeX.com u_{n + 2 =0}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a456849c03d0dac4bd5ddc8a7ede321b_l3.png)
![Rendered by QuickLaTeX.com \mathcal P_{n + 2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d7c4d78d4a74c3b4359768b595b68f7e_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N}, u_n = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ee1c16755a5ffe64963657acfa932ca0_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com \left( a , b \right) \in \mathbb{R}^2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e8a79ca8a497aecb9a93aa715c95a4d0_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com u](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png)
![Rendered by QuickLaTeX.com u_{n + 2} = 3 u_{n + 2} - 2 u_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-091301a5146500749edff258b1101e31_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e7d9750ef16217fec27533b7016511c6_l3.png)
![Rendered by QuickLaTeX.com u_0 = a](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e9ba06ae68778bc58a9d00b74de14d20_l3.png)
![Rendered by QuickLaTeX.com u_1 = b.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-dc83b3748e8fc8d8eceac55109895610_l3.png)
![Rendered by QuickLaTeX.com \varphi \left( u \right) = \left( u_0 , u_1 \right) = \left( a , b \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ebde5479ee86de54e59e259b648ff19c_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png)
![Rendered by QuickLaTeX.com \dim \left( E \right) = \dim \left( \mathbb{R}^2 \right) =2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-86c31014c4b22ac78de6837acb0d0269_l3.png)
![Rendered by QuickLaTeX.com \left( q^n \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7e7e8927de87f2da46949da824518668_l3.png)
![Rendered by QuickLaTeX.com q \in \mathbb{R}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ae4a429f324c5f42dd54acb7a5dc924b_l3.png)
![Rendered by QuickLaTeX.com E.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-108d743ccb5f048db521ce6e499f0a93_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bcc659f188465fe4226d2f333afa1af3_l3.png)
![Rendered by QuickLaTeX.com q^{n+ 2} = 3 q^{n + 1} - 2 q^n.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43681582b6000a2f01f8fd84706fad3f_l3.png)
![Rendered by QuickLaTeX.com q= 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4ed1e3c192c5d11a2ec9f39bbc0d7bb6_l3.png)
![Rendered by QuickLaTeX.com q \neq 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bb1b37c80cc835b98fd249ba99bc67b1_l3.png)
![Rendered by QuickLaTeX.com q^n,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-73959c97a600869badbe1d90915ccaa5_l3.png)
![Rendered by QuickLaTeX.com q^2 = 3q - 2,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7a48186cbff1ca79fd9c20968d86e734_l3.png)
![Rendered by QuickLaTeX.com q^2 - 3 q + 2 = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-98677ea3787f4515d72fe03760af40cf_l3.png)
![Rendered by QuickLaTeX.com q = 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-14876cb96477177fd794d1bfbbc9bde1_l3.png)
![Rendered by QuickLaTeX.com q = 2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-723b5f51c164b1b8cc8224814f28e24a_l3.png)
![Rendered by QuickLaTeX.com \left( 0 \right)_{n \in \mathbb{N}}, \left( 1 \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9e18458bde239478e0df39df1884000c_l3.png)
![Rendered by QuickLaTeX.com \left( 2^n \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-419154508cf6f9042fae823d697f5b84_l3.png)
![Rendered by QuickLaTeX.com E.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-108d743ccb5f048db521ce6e499f0a93_l3.png)
![Rendered by QuickLaTeX.com u](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png)
![Rendered by QuickLaTeX.com v](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png)
![Rendered by QuickLaTeX.com u_n = 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9d338c92ce75aed9954d1eed1a6a775c_l3.png)
![Rendered by QuickLaTeX.com v_n = 2^n.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-12b2380999b7b00266397a1ef551d6d6_l3.png)
![Rendered by QuickLaTeX.com \left( u , v \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-aa644d6ae379fe78317cb5b72449d0b7_l3.png)
![Rendered by QuickLaTeX.com a, b \in \mathbb{R}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bbf2eb02865075047f7bee20917ac312_l3.png)
![Rendered by QuickLaTeX.com a u + b v = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-35548ce7fb933ead3c41bcce4325c906_l3.png)
![Rendered by QuickLaTeX.com a= b =0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e8acbe5f49fbca3a040b55ceff141d41_l3.png)
![Rendered by QuickLaTeX.com a u + b v = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-35548ce7fb933ead3c41bcce4325c906_l3.png)
![Rendered by QuickLaTeX.com n \in \mathbb{N},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bcc659f188465fe4226d2f333afa1af3_l3.png)
![Rendered by QuickLaTeX.com a + b \times 2^n = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1e115034cdce92e9ce1b0568bcbe6f54_l3.png)
![Rendered by QuickLaTeX.com n = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b4802e0d5413bfc53b9d36f2fc992a7a_l3.png)
![Rendered by QuickLaTeX.com n= 1,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-94e46212331ca2eb61ca37210d7134fa_l3.png)
![Rendered by QuickLaTeX.com \begin{cases} a + b & = 0 \\ a + 2 b & = 0 \end{cases}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5ebb1fd226955dfbc214fe27ae72e968_l3.png)
![Rendered by QuickLaTeX.com a =b = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9632f2b660a6dc1152b2a34e29b38764_l3.png)
![Rendered by QuickLaTeX.com \left( u , v \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-aa644d6ae379fe78317cb5b72449d0b7_l3.png)
![Rendered by QuickLaTeX.com 2,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1189766c43abcab17f8edd0316219726_l3.png)
![Rendered by QuickLaTeX.com 2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cc9f98af08ef42304971ee2f5b39b7e4_l3.png)