{"id":238579,"date":"2022-05-25T17:10:58","date_gmt":"2022-05-25T15:10:58","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=238579"},"modified":"2025-09-29T13:32:17","modified_gmt":"2025-09-29T11:32:17","slug":"application-lineaire-exercice-corrige","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/application-lineaire-exercice-corrige\/","title":{"rendered":"Application lin\u00e9aire : exercice corrig\u00e9"},"content":{"rendered":"\n

Tu es en gal\u00e8re sur un exercice d’application lin\u00e9aire<\/strong> ? Pas de stress ! Gr\u00e2ce \u00e0 cet article d\u00e9di\u00e9 \u00e0 la notion : Application lin\u00e9aire : exercice corrig\u00e9<\/strong>, ce chapitre n’aura d\u00e9sormais plus aucun secret pour toi ! \u00c0 toi les bonnes notes pour tes prochaines interrogations \u00e9crites et orales sur cette notion !<\/p>\n

Et pour continuer de transformer tes difficult\u00e9s en succ\u00e8s, pense \u00e0 l’accompagnement personnalis\u00e9 d’un prof particulier de maths<\/strong><\/a>\u00a0expert en applications lin\u00e9aires. \ud83c\udf1f<\/p>\n\n\n\n

Exercice : Application lin\u00e9aire<\/h2>\n\n\n\n

\u23f0 Dur\u00e9e : 15 min<\/p>\n\n\n\n

\ud83d\udcaa Difficult\u00e9 : niveau 1\/3<\/p>\n\n\n\n

<\/p>\nDans \"\mathbb{R}^3\", on consid\u00e8re \"F=\{(x,y,z)\in\mathbb{R}^3,\;x+y=0\}\" et \"G=\mathrm{Vect}((1,0,1))\".
\n1. Interpr\u00e9ter g\u00e9om\u00e9triquement les ensembles \"F\" et \"G\".
\n2. Montrer que \"\mathbb{R}^3=F\oplus G\".
\n3. On consid\u00e8re \"p\" la projection sur \"F\" parall\u00e8lement \u00e0 \"G\". Pour tout \"(x,y,z)\in\mathbb{R}^3\", d\u00e9terminer l’expression de \"p(x,y,z)\".
\n4. On consid\u00e8re \"q\" la projection sur \"G\" parall\u00e8lement \u00e0 \"F\". Pour tout \"(x,y,z)\in\mathbb{R}^3\", d\u00e9terminer l’expression de \"q(x,y,z)\".
\n5. On consid\u00e8re \"s\" la sym\u00e9trie par rapport \u00e0 \"F\" parall\u00e8lement \u00e0 \"G\". Pour tout \"(x,y,z)\in\mathbb{R}^3\", d\u00e9terminer l’expression de \"s(x,y,z)\".
\n\n\n\n

Corrig\u00e9 de l’exercice : Application lin\u00e9aire<\/h2>\n\n\n\n

<\/p>\n1. Une base de \"G\" est \"\big((1,0,1)\big)\". Donc, \"\mathrm{dim}(G)=1\" et \"G\" est une droite vectorielle.
\nDe plus, \"(x,y,z)\in F\" si, et seulement si, \"(x,y,z)=x.(1,-1,0)+z.(0,0,1)\".
\nDonc, \"F=\mathrm{Vect}\big((1,-1,0),(0,0,1)\big)\". Les vecteurs \"(1,-1,0)\" et \"(0,0,1)\" sont non colin\u00e9aires, donc
\"\big((1,-1,0),(0,0,1)\big)\" est une base de \"F\" et \"\mathrm{dim}(F)=2\" et \"F\" est un plan vectoriel.
\n2. D’apr\u00e8s la question pr\u00e9c\u00e9dente, \"\mathrm{dim}(\mathbb{R}^3)=3=2+1=\mathrm{dim}(F)+\mathrm{dim}(G)\".
\nSoit \"(x,y,z)\in F\cap G\". Par d\u00e9finition de \"G\", il existe \"\lambda\in\mathbb{R}\" tel que \"(x,y,z)=(\lambda,0,\lambda)\".\nPuis, par d\u00e9finition de \"F\", \"z=0\". Donc, \"\lambda=0\", puis, \"(x,y,z)=(0,0,0)\".
\nOn en d\u00e9duit que \"F\cap G=\{(0,0,0)\}\" et \"\mathbb{R}^3=F\oplus G\".
\n3. Soit \"(x,y,z)\in\mathbb{R}^3\".
\nD’apr\u00e8s la question pr\u00e9c\u00e9dente, il existe \"(a,b,c)\in F\" et \"\lambda \in \mathbb{R}\" tel que \"(x,y,z)=(a,b,c)+(\lambda,0,\lambda)\in G\".
\nPar d\u00e9finition de \"p\", \"p(x,y,z)=(a,b,c)\".
\n\nOr, on a :

  <\/span>   <\/span>\"\[\left\lbrace\begin{array}{rcl}<\/p>
\nDonc, \"p(x,y,z)=(-y,y,-x-y+z)\"
\n4. D’apr\u00e8s la question pr\u00e9c\u00e9dente, pour tout \"(x,y,z)\in\mathbb{R}^3\", \"q(x,y,z)=(x+y,0,x+y)\".
\n5. On a, pour tout \"(x,y,z)\in\mathbb{R}^3\", \"s(x,y,z)=2\,p(x,y,z)-(x,y,z)=(-x-2\,y,y,-2\,x-2\,y+z)\".\n\n\n\n

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Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) \u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n\n <\/div>\n <\/section>\n<\/div>\n<\/div>\n\n\n

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\n 5\/5 - (1 vote) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Tu es en gal\u00e8re sur un exercice d’application lin\u00e9aire ? Pas de stress ! Gr\u00e2ce \u00e0 cet article (…)<\/p>\n","protected":false},"author":158,"featured_media":244760,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-238579","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/238579","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=238579"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/238579\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244760"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=238579"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=238579"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=238579"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}