{"id":237796,"date":"2022-06-02T17:13:48","date_gmt":"2022-06-02T15:13:48","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=237796"},"modified":"2024-08-28T22:47:18","modified_gmt":"2024-08-28T20:47:18","slug":"dimension-d-un-espace-vectoriel","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/dimension-d-un-espace-vectoriel\/","title":{"rendered":"Comment d\u00e9terminer la dimension d’un espace vectoriel ?"},"content":{"rendered":"\n

Tu te demandes comment d\u00e9terminer la dimension d’un espace vectoriel <\/strong>? Nous avons la r\u00e9ponse ! Voici la m\u00e9thode qui te permettra de d\u00e9terminer simplement et rapidement la dimension d’un espace vectoriel. Gr\u00e2ce aux Sherpas, obtiens 20 sur 20 \u00e0 ta prochaine interro de maths !<\/p>\n\n\n\n

Toujours perdu dans l’univers des espaces vectoriels en pr\u00e9pa scientifique ? Nos cours particuliers d’algebre<\/a> <\/strong>sont ta boussole pour naviguer dans les espaces vectoriels ! \ud83e\udded<\/p>\n\n\n

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\ud83d\udca1 Conseils m\u00e9thodologiques<\/p>\n<\/div>\n

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Pour d\u00e9terminer la dimension d’un espace vectoriel E<\/em>, on d\u00e9termine une famille B g\u00e9n\u00e9ratrice de E<\/em> (ceci montre que E<\/em> est de dimension finie), puis on v\u00e9rifie que cette famille est libre. La famille B est alors une base de E<\/em> et le nombre de vecteurs dans la famille est la dimension de E<\/em>.<\/p>\n\n <\/div>\n <\/section>\n\n\n\n

Application de la m\u00e9thode : d\u00e9terminer la dimension d’un espace vectoriel<\/strong><\/h3>\n\n\n\n
<\/div>\n\n\n\n\nOn note \"E = \mathbb{R}^3\". D\u00e9terminons la dimension de \"F = \{(x,y,z) \in E, x-y-z = 0\}\".\n
Soit \"(x,y,z) \in E\". On a \"(x,y,z) \in F\" si, et seulement si, \"(x,y,z) = (y+z, y, z) = y.(1,1,0) + z.(1,0,1)\".\n
Donc, \"F = Vect((1,1,0), (1,0,1))\". En particulier, \"F\" est un sous-espace vectoriel de \"E\".\n
Donc, la famille \"\mathcal{B} = ((1,1,0), (1,0,1))\" engendre \"F\" qui est donc de dimension finie.\n
De plus, les vecteurs \"(1,1,0)\" et \"(1,0,1)\" ne sont pas colin\u00e9aires. Donc, la famille \"\mathcal{B}\" est libre.\n
Ainsi, \"\mathcal{B}\" est une base de \"F\" et \"dim(F) = 2\".\n\n\n\n
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\"livre<\/a><\/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 <\/a><\/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 4.3\/5 - (11 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Tu te demandes comment d\u00e9terminer la dimension d’un espace vectoriel ? Nous avons la r\u00e9ponse ! Voici la (…)<\/p>\n","protected":false},"author":158,"featured_media":237857,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-237796","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\/237796","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=237796"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/237796\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/237857"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=237796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=237796"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=237796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}