{"id":277,"date":"2022-10-13T09:59:31","date_gmt":"2022-10-13T09:59:31","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=277"},"modified":"2022-10-13T09:59:46","modified_gmt":"2022-10-13T09:59:46","slug":"infinite-dimensional-vector-spaces","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2022\/10\/13\/infinite-dimensional-vector-spaces\/","title":{"rendered":"Infinite dimensional vector spaces"},"content":{"rendered":"

This is a quick note to prove that two bases of an infinite dimensional vector space have the same cardinality. We freely use the axiom of choice and other standard facts about cardinalities of infinite sets. We will in fact prove the following:<\/p>\n

Theorem<\/b>: Let \"V\" be a vector space with basis \"\{v_i\}_{i\in I}\" with \"I\" an infinite set. Let \"\{w_j\}_{j\in J}\" be a linearly independent subset of \"V\". (e.g. a basis of a subspace). Then \"|J|\leq |I|\".<\/p>\n

To prove this, WLOG \"J\" is a basis of \"V\" (by extending \"\{w_j\}_{j\in J}\" to a basis of \"V\" if necessary). For all \"i\in I\", write <\/p>\n

  <\/span>   <\/span>\"\[v_i=\sum_j c_{ij}w_j.\]\"<\/p>\n

Let \"E\subset I\times J\" be the set of pairs \"(i,j)\" with \"c_{ij}\neq 0\". Then \"E\to I\" has finite fibres, since the sum above is finite, and \"E\to J\" is surjective, since the \"v_i\" lie in the span of the \"w_j\" with \"j\" in the image of \"V\", but also the \"v_i\" generate \"V\". Since \"I\" is assumed infinite, this is enough to prove that \"|I|\geq |J|\", as required.<\/p>\n","protected":false},"excerpt":{"rendered":"

This is a quick note to prove that two bases of an infinite dimensional vector space have the same cardinality. We freely use the axiom of choice and other standard facts about cardinalities of infinite sets. We will in fact … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-277","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":false,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-4t","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/277","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=277"}],"version-history":[{"count":3,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/277\/revisions"}],"predecessor-version":[{"id":280,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/277\/revisions\/280"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=277"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=277"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=277"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}