{"id":894,"date":"2021-11-27T22:48:20","date_gmt":"2021-11-27T22:48:20","guid":{"rendered":"https:\/\/blogs.ugto.mx\/rea\/?p=894"},"modified":"2022-02-08T20:45:54","modified_gmt":"2022-02-08T20:45:54","slug":"clase-digital-15-conjuntos-generadores","status":"publish","type":"post","link":"https:\/\/blogs.ugto.mx\/rea\/clase-digital-15-conjuntos-generadores\/","title":{"rendered":"Clase digital 15: Conjuntos generadores"},"content":{"rendered":"\n\n\n<div class=\"wp-block-cover\" style=\"min-height:284px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"has-background-dim-40 wp-block-cover__gradient-background has-background-dim\"><\/span><img decoding=\"async\" class=\"wp-block-cover__image-background wp-image-895\" alt=\"\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1.jpg\" style=\"object-position:49% 67%\" data-object-fit=\"cover\" data-object-position=\"49% 67%\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"1067\" class=\"wp-block-cover__image-background wp-image-895\" alt=\"\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1.jpg\" style=\"object-position:49% 67%\" data-object-fit=\"cover\" data-object-position=\"49% 67%\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1.jpg 1600w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1-300x200.jpg 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1-1024x683.jpg 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1-768x512.jpg 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1-1536x1024.jpg 1536w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/azrbfoxp_3i-1-272x182.jpg 272w\" sizes=\"auto, (max-width: 1600px) 100vw, 1600px\" \/><\/noscript><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-base-3-color has-text-color has-large-font-size wp-block-paragraph\">Conjuntos generadores<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"introduccion\">Introducci\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00a1Hola!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00a1Qu\u00e9 bueno es encontrarte nuevamente! Te doy la bienvenida a la clase digital 15 del curso <strong>\u00c1lgebra Lineal<\/strong> en la que estudiaremos el tema Conjuntos generadores.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Para esta sesi\u00f3n ser\u00e1 necesario recordar sobre combinaciones lineales.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Veremos que a partir de algunos vectores es posible obtener otros m\u00e1s mediante suma de vectores o multiplicaci\u00f3n por escalar.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sin m\u00e1s que agregar, te invito a comenzar la clase.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"desarrollo-del-tema\">Desarrollo del tema <\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>Los vectores que forman a otros vectores de su mismo <strong>espacio vectorial <\/strong>mediante <strong>combinaciones lineales<\/strong> son <strong>conjuntos generadores<\/strong>.<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.41.41.png\" alt=\"\" class=\"wp-image-896\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"177\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.41.41.png\" alt=\"\" class=\"wp-image-896\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.41.41.png 700w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.41.41-300x76.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ejemplo 1<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Si queremos representar vectores 2 x 1 <\/strong>(dos renglones por una columna) <strong>como una combinaci\u00f3n lineal de los vectores \u201ci\u201d y \u201cj\u201d<\/strong>:<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.42.25.png\" alt=\"\" class=\"wp-image-897\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"734\" height=\"488\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.42.25.png\" alt=\"\" class=\"wp-image-897\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.42.25.png 734w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.42.25-300x199.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.42.25-272x182.png 272w\" sizes=\"auto, (max-width: 734px) 100vw, 734px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ejemplo 2<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Si queremos representar vectores 2 x1 <\/strong>(dos renglones por una columna) <strong>como una combinaci\u00f3n lineal de los vectores \u201ch\u201d y \u201ck\u201d<\/strong>:<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.50.28.png\" alt=\"\" class=\"wp-image-902\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"760\" height=\"223\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.50.28.png\" alt=\"\" class=\"wp-image-902\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.50.28.png 760w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.50.28-300x88.png 300w\" sizes=\"auto, (max-width: 760px) 100vw, 760px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ejemplo 3 <\/strong><strong><em>(obtenido del libro de Grossman)<\/em><\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Si poseemos el vector rengl\u00f3n <strong>v1 (2, -1, 4) <\/strong>y el vector <strong>v2 (4, 1, 6)<\/strong>, que forman al espacio vectorial H, es decir <strong>H = gen{v1,v2}<\/strong>,<strong> <\/strong>o bien,&nbsp;<\/li><\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">H = gen{(2,-1,4),(4,1,6)}, \u00bfCu\u00e1l es la apariencia de \u201cH\u201d?<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Para resolver la pregunta, primero seleccionamos un vector inc\u00f3gnita del espacio vectorial H, por ejemplo v = (x, y, z).&nbsp;<\/li><li>Ahora v debe formarse como combinaci\u00f3n lineal de <strong>v1 (2, -1, 4) <\/strong>y del vector <strong>v2 (4, 1, 6)<\/strong>, entonces:&nbsp;<\/li><\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">v = a<sub>1<\/sub>v1 + a<sub>2<\/sub>v2<br>(x, y, z) = a<sub>1<\/sub>(2, -1, 4) + a<sub>2<\/sub>(4, 1, 6)<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>A partir de la ecuaci\u00f3n:<\/li><\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">(x, y, z) = a<sub>1<\/sub>(2, -1, 4) + a<sub>2<\/sub>(4, 1, 6)<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Obtenemos que:<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.51.35.png\" alt=\"\" class=\"wp-image-903\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"399\" height=\"142\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.51.35.png\" alt=\"\" class=\"wp-image-903\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.51.35.png 399w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.51.35-300x107.png 300w\" sizes=\"auto, (max-width: 399px) 100vw, 399px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>Resolvemos por Gauss \u2013 Jordan hasta obtener un rengl\u00f3n s\u00f3lo con ceros:<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.52.16.png\" alt=\"\" class=\"wp-image-904\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"762\" height=\"390\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.52.16.png\" alt=\"\" class=\"wp-image-904\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.52.16.png 762w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2021\/11\/Captura-de-Pantalla-2021-11-09-a-las-13.52.16-300x154.png 300w\" sizes=\"auto, (max-width: 762px) 100vw, 762px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>Es decir, el espacio vectorial H es la siguiente ecuaci\u00f3n:<\/li><\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">-5x\/3 + 2y\/3 + z = 0<br>O bien, <strong>-5x + 2y + 3z = 0<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Y su conjunto generador son el vector <strong>v1 (2, -1, 4) <\/strong>y el vector <strong>v2 (4, 1, 6)<\/strong><\/li><li>Es decir, cualquier conjunto de respuestas (vectores) que satisfaga la ecuaci\u00f3n (espacio vectorial H) se puede formar con una combinaci\u00f3n lineal de v1 y v2.&nbsp;<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>NOTAS<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>El <strong>espacio vectorial R2 <\/strong>se refiere al espacio 2D para cualquier valor de X \u00f3 Y.&nbsp;<\/li><li>Todas las ecuaciones de recta que pasen por el origen son <strong>subespacios vectoriales <\/strong>de R2, es decir, R2 contiene a todas los espacios vectoriales de dos inc\u00f3gnitas.&nbsp;<\/li><li>El <strong>espacio vectorial R3<\/strong>&nbsp; se refiere al espacio 3D, es decir, el que contiene todos los valores para X, Y \u00f3 Z.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"conclusion\">Conclusi\u00f3n <\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Concluimos la sesi\u00f3n mencionando que:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Los vectores que forman a otros vectores de su mismo <strong>espacio vectorial <\/strong>mediante <strong>combinaciones lineales<\/strong> son <strong>conjuntos generadores<\/strong>.&nbsp;<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Has concluido la clase. \u00a1Muchas felicidades! Recuerda que est\u00e1s por terminar este curso, que tu \u00e1nimo no decaiga contin\u00faa esforz\u00e1ndote. No olvides realizar la tarea asignada y mandarla como corresponde. Te espero en la pr\u00f3xima sesi\u00f3n, \u00a1hasta pronto!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"fuentes-de-informacion\">Fuentes de informaci\u00f3n <\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>Grossman, S. I. (2004). Algebra Lineal y sus Aplicaciones. (5<sup>a<\/sup> ed.). M\u00e9xico: McGraw-Hill.<\/li><li>Anton, H. (2011). Introducci\u00f3n al Algebra Lineal. (5<sup>a<\/sup> ed.). M\u00e9xico: Limusa Wiley.<\/li><li>Campbell, H. G. (1980). Linear Algebra with Applications. Atlanta: Prentice-Hall.<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introducci\u00f3n \u00a1Hola! \u00a1Qu\u00e9 bueno es encontrarte nuevamente! Te doy la bienvenida a la clase digital 15 del curso \u00c1lgebra Lineal en la que estudiaremos el tema Conjuntos generadores.&nbsp; Para esta sesi\u00f3n ser\u00e1 necesario recordar sobre combinaciones lineales.&nbsp; Veremos que a partir de algunos vectores es posible obtener otros m\u00e1s mediante suma de vectores o multiplicaci\u00f3n &#8230; <a title=\"Clase digital 15: Conjuntos generadores\" class=\"read-more\" href=\"https:\/\/blogs.ugto.mx\/rea\/clase-digital-15-conjuntos-generadores\/\" aria-label=\"Leer m\u00e1s sobre Clase digital 15: Conjuntos generadores\">Leer m\u00e1s<\/a><\/p>\n","protected":false},"author":142,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","episode_type":"","audio_file":"","podmotor_file_id":"","podmotor_episode_id":"","cover_image":"","cover_image_id":"","duration":"","filesize":"","filesize_raw":"","date_recorded":"","explicit":"","block":"","itunes_episode_number":"","itunes_title":"","itunes_season_number":"","itunes_episode_type":"","footnotes":""},"categories":[10,11],"tags":[41,61,62],"class_list":["post-894","post","type-post","status-publish","format-standard","hentry","category-ingenieria-civil","category-uda-algebra-lineal","tag-clase-digital","tag-jorge-arturo-gutierrez-camarena","tag-neli04035"],"acf":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/894","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/users\/142"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/comments?post=894"}],"version-history":[{"count":3,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/894\/revisions"}],"predecessor-version":[{"id":7295,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/894\/revisions\/7295"}],"wp:attachment":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/media?parent=894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/categories?post=894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/tags?post=894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}