{"id":565,"date":"2025-11-28T08:21:44","date_gmt":"2025-11-28T07:21:44","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=565"},"modified":"2025-11-28T08:41:52","modified_gmt":"2025-11-28T07:41:52","slug":"alg-ortogonalizacion-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=565","title":{"rendered":"ALG: Ortogonalizaci\u00f3n con maxima"},"content":{"rendered":"<p>Abordemos una de los procesos m\u00e1s importantes en este tema:<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Dar una base ortogonal de la variedad \\(S=\\left\\{\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix}+\\left.\\begin{bmatrix}a+b&#038;3a-b\\\\ b&#038; -a\\end{bmatrix}\\right|a,b\\in\\mathbb{R}\\right\\}\\) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1() {\n  var htmlShow1 = document.getElementById(\"html-show1\");\n  if (htmlShow1.style.display === \"none\") {\n    htmlShow1.style.display = \"block\";\n  } else {\n    htmlShow1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGbaseorto01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Cu\u00e1l ser\u00eda la traza de la matriz producto de una base ortogonal obtenida de las matrices: \\(\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix}\\) y \\(\\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3\\end{bmatrix}\\) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2() {\n  var htmlShow2 = document.getElementById(\"html-show2\");\n  if (htmlShow2.style.display === \"none\") {\n    htmlShow2.style.display = \"block\";\n  } else {\n    htmlShow2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGbaseorto02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Cu\u00e1l ser\u00eda la norma de la suma de los vectores de una base ortogonal obtenida de los polinomios: \\(\\left( -3 {{x}^{2}}+2 x+1\\right)\\) y\\(\\left( {{x}^{2}}-x-2\\right)\\) en \\(\\mathbb{R}_2[x]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3() {\n  var htmlShow3 = document.getElementById(\"html-show3\");\n  if (htmlShow3.style.display === \"none\") {\n    htmlShow3.style.display = \"block\";\n  } else {\n    htmlShow3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGbaseorto03.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Sea B={(2,1,1),(1,0,10),(2,-3,11)} una base de \\(\\mathbb{R}^3\\), \u00bfcu\u00e1l es el la suma de las normas al cuadrado de una base ortogonal obtenida por un proceso de ortogonalizaci\u00f3n de Gram\u2013Schmidt? <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/2021\/12\/Ejer_ortogonalizacion_GramSchmidt.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<h2>Complemento Ortogonal<\/h2>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(\\pi:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+y-z=0,\\ x-y+3t=0\\}\\) un plano en \\(\\mathbb{R}^4\\) y \\(u\\):[\\(a\\), 3, -2, -3]. \u00bfCu\u00e1l es el valor de \\(a\\) para  que \\(u\\in S^\\perp\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv25() {\n  var htmlShow25 = document.getElementById(\"html-show25\");\n  if (htmlShow25.style.display === \"none\") {\n    htmlShow25.style.display = \"block\";\n  } else {\n    htmlShow25.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv25()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show25\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGorto03b.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(S=\\{[[3a+2b,-2a-b],[b,a]]\\in \\mathcal{M}_2(\\mathbb{R})\\}\\). Sean \\(x,y\\in\\mathbb{R}\\), tales que  \\(A=[[2,x],[y,-2]]\\in S^\\bot\\), \u00bfcu\u00e1l es \\(\\|A\\|^2\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv35() {\n  var htmlShow35 = document.getElementById(\"html-show35\");\n  if (htmlShow35.style.display === \"none\") {\n    htmlShow35.style.display = \"block\";\n  } else {\n    htmlShow35.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv35()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show35\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGorto04.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(\\pi:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+3y-z=0,\\ y+2z-t=0\\}\\). \u00bfCu\u00e1l de los vectores a:[8,13,-2,-1], b:[8,-13,2,-1] y c:[-8,13,-2,1], pertenece a su ortogonal? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv45() {\n  var htmlShow45 = document.getElementById(\"html-show45\");\n  if (htmlShow45.style.display === \"none\") {\n    htmlShow45.style.display = \"block\";\n  } else {\n    htmlShow45.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv45()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show45\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGorto05.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(S:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+y-z=0,\\ x-y+3t=0\\}\\). \u00bfCu\u00e1l es la \\(\\|\\textbf{proy}_S([-1,0,2,1])\\|\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv29() {\n  var htmlShow29 = document.getElementById(\"html-show29\");\n  if (htmlShow29.style.display === \"none\") {\n    htmlShow29.style.display = \"block\";\n  } else {\n    htmlShow29.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv29()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show29\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGproy02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>Abordemos una de los procesos m\u00e1s importantes en este tema: Ejemplo: Dar una base ortogonal de la variedad \\(S=\\left\\{\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix}+\\left.\\begin{bmatrix}a+b&#038;3a-b\\\\ b&#038; -a\\end{bmatrix}\\right|a,b\\in\\mathbb{R}\\right\\}\\) Soluci\u00f3n: Ejemplo: Cu\u00e1l ser\u00eda la traza de la matriz&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[7],"class_list":["post-565","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/565","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=565"}],"version-history":[{"count":1,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/565\/revisions"}],"predecessor-version":[{"id":566,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/565\/revisions\/566"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=565"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=565"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}