{"id":234847,"date":"2022-07-08T17:35:56","date_gmt":"2022-07-08T15:35:56","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=234847"},"modified":"2025-09-29T13:35:57","modified_gmt":"2025-09-29T11:35:57","slug":"matrice-et-application-lineaire-exercice-corrige","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/matrice-et-application-lineaire-exercice-corrige\/","title":{"rendered":"Matrice et application lin\u00e9aire exercice corrig\u00e9"},"content":{"rendered":"\n

Vous \u00e9tudiez actuellement la notion de matrice et application lin\u00e9aire<\/strong> ? Rassurez-vous, gr\u00e2ce \u00e0 l’article matrice et application lin\u00e9aire exercice corrig\u00e9<\/strong>, vous allez pouvoir ma\u00eetriser cette notion sur le bout des doigts gr\u00e2ce \u00e0 des m\u00e9thodologies abouties !<\/p>\n

Pour approfondir encore votre compr\u00e9hension, d\u00e9couvrez nos cours particuliers d’alg\u00e8bre<\/a> qui vous aideront \u00e0 ma\u00eetriser pleinement les matrices et applications lin\u00e9aires. \ud83d\udcda<\/p>\n\n\n\n\n\n

Exercice : Matrice et application lin\u00e9aire<\/h2>\n\n\n\n

<\/p>\nSoit

  <\/span>   <\/span>\"\[A=\begin{pmatrix}<\/p> \"\in \mathcal{M}_2(\mathbb{K})\". On consid\u00e8re \"f:\begin{array}[t]{rcl} \mathcal{M}_2(\mathbb{K}) & \to & \mathcal{M}_2(\mathbb{K})\\\n

  • Montrer que \"f \in \mathcal{L}\big(\mathcal{M}_2(\mathbb{R})\big)\". <\/li>\n
  • D\u00e9terminer la matrice de \"f\" dans la base canonique \"\mathcal{B}=\big(E_{1,1},E_{2,1},E_{1,2},E_{2,2}\big)\" de \"\mathcal{M}_2(\mathbb{K})\". <\/li>\n
  • D\u00e9terminer la trace de \"f\". <\/li>\n\n\n\n

    Corrig\u00e9 de l’exercice <\/h2>\n\n\n\n

    <\/p>\n\\begin{enumerate}\n

  • Soient \"(M,N)\in \mathcal{M}_2(\mathbb{K})\times \mathcal{M}_2(\mathbb{K})\" et \"\lambda \in\mathbb{K}\". On a :\n

      <\/span>   <\/span>\"\[f(\lambda.M+N)=A\times(\lambda.M+N)=\lambda.A M+ A N=\lambda.f(M)+f(N).\]\"<\/p> <\/li>\n

  • Donc, \"f\" est lin\u00e9aire. De plus, le produit de matrices de \"\mathcal{M}_2(\mathbb{K})\" appartient \u00e0 \"\mathcal{M}_2(\mathbb{K})\".
    \nDonc, \"f \in \mathcal{L}\big(\mathcal{M}_2(\mathbb{K})\big)\".\n<\/li>
  • On a :\n

      <\/span>   <\/span>\"\[f(E_{1,1})=\begin{pmatrix}<\/p>,\n

      <\/span>   <\/span>\"\[f(E_{2,1})=\begin{pmatrix}<\/p>,\n

      <\/span>   <\/span>\"\[f(E_{1,2})=\begin{pmatrix}<\/p>\n et\n

      <\/span>   <\/span>\"\[f(E_{2,2})=\begin{pmatrix}<\/p> Donc,\n

      <\/span>   <\/span>\"\[A = \begin{pmatrix}<\/p> <\/li>\n

  • On a \"\tr(f)=\tr(A)=4\". <\/li>\n\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) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n

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

    Vous \u00e9tudiez actuellement la notion de matrice et application lin\u00e9aire ? Rassurez-vous, gr\u00e2ce \u00e0 l’article matrice et application lin\u00e9aire (…)<\/p>\n","protected":false},"author":158,"featured_media":244590,"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-234847","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\/234847","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=234847"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234847\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244590"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=234847"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=234847"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=234847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}