{"id":237862,"date":"2022-06-02T17:12:27","date_gmt":"2022-06-02T15:12:27","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=237862"},"modified":"2025-09-29T11:33:02","modified_gmt":"2025-09-29T09:33:02","slug":"application-lineaire","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/application-lineaire\/","title":{"rendered":"Qu\u2019est-ce qu\u2019une application lin\u00e9aire ?"},"content":{"rendered":"\n
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Tu te demandes ce qu’est une application lin\u00e9aire <\/strong>? Quelle est sa d\u00e9finition, et comment montrer qu’une application est lin\u00e9aire<\/strong> ? Apr\u00e8s avoir lu ce cours consacr\u00e9 \u00e0 l’application lin\u00e9aire, tu auras la r\u00e9ponse \u00e0 toutes ces questions. \u00c0 toi le 20 sur 20 \u00e0 la prochaine interro de maths !<\/p>\n\n\n\n

Pour explorer plus en profondeur le r\u00f4le fondamental des applications lin\u00e9aires dans l’espace vectoriel<\/strong>, nos cours particuliers d’alg\u00e8bre<\/a><\/strong> sont une ressource pr\u00e9cieuse et accessible.<\/p>\n\n\n\n

<\/div>\n\n\n\n\nDans ce chapitre, \"E\" , \"F\" et \"G\" d\u00e9signent trois \"\mathbb{K}\"-espace vectoriel.\n\n\n\n

\ud83d\udccdD\u00e9finition<\/span> : Application lin\u00e9aire<\/strong><\/p>\n\n\n\n\nSoit \"f : E \rightarrow F\" une application. On dit que \"f\" est lin\u00e9aire lorsque :\n
(i) \"\forall (u,v) \in E^2\", \"f(u + v) = f(u) + f(v)\"\n
(ii) \"\forall \lambda \in \mathbb{K}\", \"\forall u \in E\", \"f(\lambda.u) = \lambda.f(u)\".\n
On note \"\mathcal{L}(E,F)\" l’ensemble des applications lin\u00e9aires de \"E\" dans \"F\".\n\n\n\n

Remarques :<\/strong><\/p>\n\n\n\n\n

  • Une application \"f\" de \"E\" dans \"F\" est lin\u00e9aire si, et seulement si, \n

      <\/span>   <\/span>\"\[<\/p>\nC’est cette caract\u00e9risation qui est utilis\u00e9e en pratique pour montrer qu’une application est lin\u00e9aire. <\/li>\n

  • On montre par r\u00e9currence sur \"p \in \mathbb{N}^*\" que, pour tout \"(\lambda_1,...,\lambda_p) \in \mathbb{K}^p\" et pour tout \"(u_1,...,u_p) \in E^p\", on a : <\/li>\n

      <\/span>   <\/span>\"\[f(\sum_{k=1}^p \lambda_k.u_k) = \sum_{k=1}^p \lambda_k.f(u_k)$.\]\"<\/p>\n\n\n\n

    Exemple :<\/strong><\/p>\n\n\n\n\nL’application \"\begin{array}{rcl} est lin\u00e9aire.\n\n
    Plus g\u00e9n\u00e9ralement, pour tout \"\lambda \in \mathbb{K}\", l’application \"\lambda.\text{Id}_E\" est lin\u00e9aire.\n\n\n\n

    Remarques :<\/strong><\/p>\n\n\n\n\nSoit \"f\" une application lin\u00e9aire de \"E\" dans \"F\".\n

  • On a : \"f(0_E) = 0_F\". En effet : \n

      <\/span>   <\/span>\"\[f(0_E) = f(0_E + 0_E) = f(0_E) + f(0_E) .\]\"<\/p>\nEn ajoutant \"-f(0_E)\" \u00e0 chaque membre de l’\u00e9galit\u00e9, on a \"f(0_E) = 0_F\".<\/li>\n

  • En particulier, si \"f(0_E) \ne 0_F\", alors \"f\" n\u2019est pas lin\u00e9aire.<\/li>\n
  • On a : pour tout \"x \in E\", \"f(-x) = -f(x)\". En effet, il suffit de prendre \"\lambda = -1\", \"u = x\" et \"v = 0_E\" dans la d\u00e9finition d’application lin\u00e9aire.<\/li>\n\n\n\n
    <\/div>\n\n\n\n

    \ud83d\udca1 Conseils m\u00e9thodologiques<\/span> : Montrer qu’une application est lin\u00e9aire<\/strong><\/p>\n\n\n\n\nPour montrer qu’une application \"f : E \to F\" est lin\u00e9aire, on fixe \"\lambda \in \mathbb{K}\" et \"(u,v) \in E^2\" et on montre que \"f(\lambda.u+v) = \lambda.f(u) + f(v)\" en utilisant la d\u00e9finition de \"f\".\n\n\n\n

    Exemple<\/strong> :<\/strong><\/p>\n\n\n\n\nL’application \"\begin{array}{rcl} est lin\u00e9aire.\n\n\"\varphi : \mathcal{C}^1(\mathbb{R},\mathbb{R}) \to \mathcal{C}^0(\mathbb{R},\mathbb{R})\" est une application lin\u00e9aire. En effet : Soient \"(f,g) \in \mathcal{C}^1(\mathbb{R},\mathbb{R}) \times \mathcal{C}^1(\mathbb{R},\mathbb{R})\" et \"\lambda \in \mathbb{R}\". On a : \n

      <\/span>   <\/span>\"\[\varphi(\lambda.f+g) = (\lambda.f+g)' = \lambda.f' + g' = \lambda.\varphi(f) + \varphi(g).\]\"<\/p>\n\"\varphi\" est une application lin\u00e9aire.\n\n\n\n

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    \"livre<\/figure>\n<\/div>\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 te demandes ce qu\u2019est une application lin\u00e9aire ? Quelle est sa d\u00e9finition, et comment montrer qu\u2019une application (…)<\/p>\n","protected":false},"author":158,"featured_media":237946,"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-237862","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\/237862","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=237862"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/237862\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/237946"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=237862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=237862"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=237862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}