{"id":239569,"date":"2022-05-10T17:08:55","date_gmt":"2022-05-10T15:08:55","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=239569"},"modified":"2025-09-29T11:26:39","modified_gmt":"2025-09-29T09:26:39","slug":"tout-savoir-sur-le-systeme-lineaire","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/tout-savoir-sur-le-systeme-lineaire\/","title":{"rendered":"Tout savoir sur le syst\u00e8me lin\u00e9aire"},"content":{"rendered":"\n

Vous avez du mal avec la notion de syst\u00e8me lin\u00e9aire<\/strong> ? Pas d’inqui\u00e9tude ! Gr\u00e2ce \u00e0 ce cours d\u00e9di\u00e9 \u00e0 la notion de syst\u00e8me lin\u00e9aire<\/strong>, familiarisez-vous davantage avec des m\u00e9thodologies bien structur\u00e9es qui vous permettront de d\u00e9crocher de bonnes notes \u00e0 vos prochaines interrogations orales et \u00e9crites !<\/p>\n\n\n\n

Et si les inconnues et les rangs de matrices te semblent encore insurmontables, un prof particulier de maths<\/a><\/strong> peut transformer ce d\u00e9fi en une victoire math\u00e9matique. \ud83c\udfc6<\/p>\n\n\n\n

Syst\u00e8mes lin\u00e9aires<\/h2>\n\n\n\n

Proposition<\/h3>\n\n\n\n

<\/p>\nL’ensemble des solutions d’un syst\u00e8me lin\u00e9aire homog\u00e8ne \u00e0 \"n\" \u00e9quations et \"p\" inconnues est le noyau de la matrice des coefficients du syst\u00e8me.\n\n\n\n

D\u00e9monstration<\/h4>\n\n\n\n

<\/p>\nOn consid\u00e8re le syst\u00e8me lin\u00e9aire homog\u00e8ne \u00e0 \"n\" \u00e9quations et \"p\" inconnues :\n

  <\/span>   <\/span>\"\[(\mathcal{S})\, \left\lbrace<\/p>\nOn note \"A=(a_{i,j})_{\substack{1\leq i\leq n\\ 1\leq j\leq p}}\" la matrice associ\u00e9e au syst\u00e8me. Par d\u00e9finition, \"(x_1,\dots,x_p)\" est solution de \"(\mathcal{S})\" si, et seulement si, \"A \begin{pmatrix}. Autrement dit, l’ensemble des solutions de \"(\mathcal{S})\" est \"\mathrm{Ker}(A)\".\n\n\n\n

D\u00e9finition : Rang d’un syst\u00e8me<\/h3>\n\n\n\n

Le rang d’un syst\u00e8me lin\u00e9aire homog\u00e8ne est le rang de la matrice de ses coefficients.<\/p>\n\n\n\n

Proposition<\/h3>\n\n\n\n

<\/p>\nLa dimension de l’ensemble des solutions d’un syst\u00e8me lin\u00e9aire homog\u00e8ne \u00e0 \"n\" \u00e9quations et \"p\" inconnues de rang \"r\" est \u00e9gal \u00e0 \"p-r\".\n\n\n\n

D\u00e9monstration<\/h4>\n\n\n\n

<\/p>\nOn note \"A\in\mathcal{M}_{n,p}(\mathbb{K})\" la matrice des coefficients du syst\u00e8mes. On sait que \"\mathrm{dim}(\mathrm{Ker}(A))=p-\mathrm{rg}(A)\".
\nOr, \"\mathrm{Ker}(A)\" est l’ensemble des solutions du syst\u00e8me.\n\n\n\n

Proposition<\/h3>\n\n\n\n

<\/p>\nOn consid\u00e8re un syst\u00e8me lin\u00e9aire dont l’\u00e9criture matricielle est \"A\times X=B\" o\u00f9 \"A\in\mathcal{M}_{n,p}(\)mathbb{K}\", \"X\in\mathcal{M}_{p,1}(\mathbb{K})\" et \"B\in\mathcal{M}_{n,1}(\mathbb{K})\".
\nLe syst\u00e8me \"A X=B\" est compatible si, et seulement si \"B\in\mathrm{Im}(A)\".\n\n\n\n

D\u00e9monstration<\/h4>\n\n\n\n

<\/p>\nCons\u00e9quence imm\u00e9diate de la d\u00e9finition de \"\mathrm{Im}(A)\".\n\n\n\n

Remarque : Structure affine de l’ensemble des solutions d’un syst\u00e8me lin\u00e9aire<\/h4>\n\n\n\n

<\/p>\nSoit \"A\times X=B\" un syst\u00e8me compatible. On note \"X_0\" une solution particuli\u00e8re du syst\u00e8me.
\nUn vecteur \"X\" est solution du syst\u00e8me si, et seulement si, \"AX=B\".
\nOr, \"AX_0=B\". Donc, \"X\" est solution du syst\u00e8me si, et seulement si, \"X-X_0\in\mathrm{Ker}(A)\".
\nOn en d\u00e9duit que l’ensemble des solutions du syst\u00e8me est le sous-espace affine de \"\K^p\" dirig\u00e9 par \"\mathrm{Ker}(A)\" et passant pas \"X_0\".\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":"

Vous avez du mal avec la notion de syst\u00e8me lin\u00e9aire ? Pas d’inqui\u00e9tude ! Gr\u00e2ce \u00e0 ce cours (…)<\/p>\n","protected":false},"author":158,"featured_media":165972,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[770,78,345],"class_list":["post-239569","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-fiche-revision","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/239569","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=239569"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/239569\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/165972"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=239569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=239569"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=239569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}