{"id":96,"date":"2026-09-21T08:25:39","date_gmt":"2026-09-21T06:25:39","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=96"},"modified":"2026-08-20T12:57:55","modified_gmt":"2026-08-20T10:57:55","slug":"alg-vectores-y-matrices-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=96","title":{"rendered":"ALG: Vectores y matrices con Maxima"},"content":{"rendered":"<h2>Maxima<\/h2>\n<p>Las pr\u00e1cticas la realizaremos con Maxima, un sistema para la manipulaci\u00f3n de expresiones simb\u00f3licas y num\u00e9ricas. Esta herramienta inform\u00e1tica  nos ayudar\u00e1 a resolver problemas de la asignatura de forma sencilla y aplicada.<\/p>\n<p>El sitio de descargas est\u00e1 en siguiente enlace <a href=\"https:\/\/maxima.sourceforge.io\/es\/download.html\" target=\"_blank\" rel=\"noopener\">descarga maxima<\/a>. Aunque pod\u00e9is trabajar con \u00e9l, hay un entorno, muy intuitivo y c\u00f3modo que ser\u00e1 con el que trabajaremos <a href=\"http:\/\/wxmaxima-developers.github.io\/wxmaxima\/\" target=\"_blank\" rel=\"noopener\">wxMaxima<\/a> (si ten\u00e9is problemas con esa versi\u00f3n, hay una versi\u00f3n para windows en <a href=\"https:\/\/wolfgang.dautermann.at\/maxima\/nightlybuild\/\" rel=\"noopener\" target=\"_blank\">https:\/\/wolfgang.dautermann.at\/maxima\/nightlybuild\/<\/a>), con una colecci\u00f3n amplia de men\u00fas en los que, agrupados por tem\u00e1ticas, se recogen las operaciones m\u00e1s habituales, de suerte que el usuario principiante no necesita recordar los comandos, aunque os aconsejo tenerlos a mano. Con wxMaxima podremos resolver nuestras pr\u00e1cticas y presentar los resultados de forma m\u00e1s sencilla.<\/p>\n<figure style=\"width: 459px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" fetchpriority=\"high\" class=\"\" src=\"http:\/\/wxmaxima-developers.github.io\/wxmaxima\/images\/linux_5.png\" alt=\"interface for the computer algebra system Maxima\" width=\"459\" height=\"269\" \/><figcaption class=\"wp-caption-text\">wxMaxima is a document based interface for the computer algebra system Maxima. wxMaxima provides menus and dialogs for many common maxima commands, autocompletion, inline plots and simple animations. wxMaxima is distributed under the GPL license.<\/figcaption><\/figure>\n<p>En la web de maxima dispon\u00e9is de documentaci\u00f3n, <a href=\"https:\/\/maxima.sourceforge.io\/docs\/manual\/es\/maxima.html\" target=\"_blank\">https:\/\/maxima.sourceforge.io\/docs\/manual\/es\/maxima.html<\/a>. Nosotros no veremos todos los comandos y funciones, solo utilizaremos los que mostremos en los ejercicios. Tambi\u00e9n, encontrar\u00e9is bastante material, aparte del que os proporcionaremos, como el del profesor Jeronimo Alaminos et al, <a href=\"http:\/\/www.ugr.es\/~fjperez\/textos\/Practicas%20con%20wxMaxima.pdf\" target=\"_blank\" rel=\"noopener\">Pr\u00e1cticas de ordenador con Maxima<\/a>, un valioso documento para la iniciaci\u00f3n. O la web del excelente profesor Jos\u00e9 Manuel Mira Ros(del que tengo un grato recuerdo) con su <a href=\"https:\/\/webs.um.es\/mira\/maxima\/manualico.php\" target=\"_blank\" rel=\"noopener\">manualico<\/a>, que os ser\u00e1 de tremenda ayuda.<\/p>\n<hr \/>\n<h2>Pr\u00e1cticas con Maxima<\/h2>\n<p>Hoy nos iniciamos en un sistema para la manipulaci\u00f3n de expresiones simb\u00f3licas y num\u00e9ricas, Maxima. Un herramienta inform\u00e1tica que nos ayudar\u00e1 a resolver problemas de la asignatura de forma sencilla y aplicada.<\/p>\n<p>Para comenzar, nos iniciaremos en la definici\u00f3n de vectores y matrices, y las operaciones que podemos hacer con ellos.<\/p>\n<h3>Vectores<\/h3>\n<p>Un vector se define utilizando [] y los elementos del vector separados por comas. Con los vectores podemos hacer las operaciones b\u00e1sicas de suma y multiplicaci\u00f3n por escalar.<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sean los vectores \\(v:[1,2,3]\\) y \\(u:[4,5,6]\\), calcular \\(2v-3u\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31() {\n  var htmlShow31 = document.getElementById(\"html-show31\");\n  if (htmlShow31.style.display === \"none\") {\n    htmlShow31.style.display = \"block\";\n  } else {\n    htmlShow31.style.display = \"none\";\n  }\n}\n<\/script><button onclick=\"showHtmlDiv31()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i3)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">5<\/span>,<span class=\"code_number\">6<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_number\">2<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">;<\/span> <\/span> <\/td>\n<\/tr>\n<\/table>\n<p>    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o3) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mo>[<\/mo>\n            <mi>\u2212<\/mi>\n            <mn>10<\/mn>\n            <mo>,<\/mo>\n            <mi>\u2212<\/mi>\n            <mn>11<\/mn>\n            <mo>,<\/mo>\n            <mi>\u2212<\/mi>\n            <mn>12<\/mn>\n            <mo>]<\/mo>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math>\n<\/div>\n<hr \/>\n<p>Como hemos visto en el ejercicio anterior, el operador &quot;<strong>+<\/strong>&quot; suma componente a componente de dos vectores. El operador &quot;<strong>*<\/strong>&quot; multiplica un escalar por cada uno de los elementos de un vector. Una diferencia cuando utilizamos los operadores &quot;<strong>*<\/strong>&quot; o &quot;<strong>.<\/strong>&quot; entre vectores. El operador &quot;<strong>*<\/strong>&quot; entre dos vectores nos proporciona otro vector de la misma dimensi\u00f3n donde las componentes se han multiplicado dos a dos; mientras que el operador &quot;<strong>.<\/strong>&quot; nos proporciona el producto escalar de dos vectores.<\/p>\n<p>Otro operador importante es &quot;<strong>^<\/strong>&quot;, en este caso \\(v\\)<em>^n<\/em>, siendo \\(v\\) un vector, eleva cada una de las componentes de \\(v\\) a \\(n\\). <\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Como veremos m\u00e1s adelante, definimos el producto escalar, eucl\u00eddeo, de los vectores \\(v:[v_1,v_2,\\ldots,v_n]\\) y \\(u:[u_1,u_2,\\ldots,u_n]\\), como \\[v\\bullet u=\\sum_{i=1}^nv_i\\cdot u_i.\\]<br \/>\nConstruir una funci\u00f3n que dados dos vectores nos devuelva su producto escalar.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31b() {\n  var htmlShow31b = document.getElementById(\"html-show31b\");\n  if (htmlShow31b.style.display === \"none\") {\n    htmlShow31b.style.display = \"block\";\n  } else {\n    htmlShow31b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv31b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31b\" style=\"display: none;\">\n <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">p_escalar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Veamos un ejemplo:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">p_escalar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }22\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dados los vectores \\(u\\):[2,-2], \\(v\\):[5,8] y \\(w\\):[-4,3], y el producto escalar eucl\u00eddeo, \u00bfcu\u00e1l es el valor de \\(u\\bullet (v-2w)\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31b5() {\n  var htmlShow31b5 = document.getElementById(\"html-show31b5\");\n  if (htmlShow31b5.style.display === \"none\") {\n    htmlShow31b5.style.display = \"block\";\n  } else {\n    htmlShow31b5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv31b5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31b5\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">p_escalar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">p_escalar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[22\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Como veremos m\u00e1s adelante, definimos la norma, eucl\u00eddea, de un vector \\(v:[v_1,v_2,\\ldots,v_n]\\)  como \\[\\|v\\|=\\sqrt{\\sum_{i=1}^nv_i^2}.\\]<br \/>\nConstruir una funci\u00f3n que dado un vector nos devuelva su norma.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31b2() {\n  var htmlShow31b2 = document.getElementById(\"html-show31b2\");\n  if (htmlShow31b2.style.display === \"none\") {\n    htmlShow31b2.style.display = \"block\";\n  } else {\n    htmlShow31b2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv31b2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31b2\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">norma<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Veamos un ejemplo:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">norma<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">norma<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\sqrt{30}\\]<\/p>\n<p>\\[\\operatorname{ }5.477225575051661\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Continuando con los ejemplos anteriores, llamamos normalizar un vector al proceso de dividir un vector por su norma. Construir una funci\u00f3n que, dado un vector, nos lo devuelva normalizado.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3t() {\n  var htmlShow3t = document.getElementById(\"html-show3t\");\n  if (htmlShow3t.style.display === \"none\") {\n    htmlShow3t.style.display = \"block\";\n  } else {\n    htmlShow3t.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3t()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3t\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">norma<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">normalizar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">norma<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">normalizar<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\frac{1}{\\sqrt{30}},\\frac{2}{\\sqrt{30}},\\frac{3}{\\sqrt{30}},\\frac{4}{\\sqrt{30}}\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<h2>Matrices<\/h2>\n<p>Si queremos utilizar matrices nos bastar\u00e1 con definirla mediante <strong>matrix()<\/strong>. Las filas de definimos como vectores:<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Definir las matrices  \\(A=\\begin{bmatrix}1 &#038; 2 &#038; 3\\\\<br \/>\n 4 &#038; 5 &#038; 6\\end{bmatrix}\\) y \\(B=\\begin{bmatrix}1 &#038; 2 \\\\ 3&#038; 4 \\\\ 5 &#038; 6\\end{bmatrix}\\). <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv32() {\n  var htmlShow32 = document.getElementById(\"html-show32\");\n  if (htmlShow32.style.display === \"none\") {\n    htmlShow32.style.display = \"block\";\n  } else {\n    htmlShow32.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv32()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show32\" style=\"display: none;\">\n   <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i2)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n          <span class=\"input\">           <span class=\"code_variable\">A<\/span>            <span class=\"code_operator\">:<\/span>            <span class=\"code_function\">matrix<\/span>([            <span class=\"code_number\">1<\/span>,            <span class=\"code_number\">2<\/span>,            <span class=\"code_number\">3<\/span>],[            <span class=\"code_number\">4<\/span>,            <span class=\"code_number\">5<\/span>,            <span class=\"code_number\">6<\/span>])            <span class=\"code_endofline\">;<\/span>            <br \/>            <span class=\"code_variable\">B<\/span>            <span class=\"code_operator\">:<\/span>            <span class=\"code_function\">matrix<\/span>([            <span class=\"code_number\">1<\/span>,            <span class=\"code_number\">2<\/span>],[            <span class=\"code_number\">3<\/span>,            <span class=\"code_number\">4<\/span>],[            <span class=\"code_number\">5<\/span>,            <span class=\"code_number\">6<\/span>])            <span class=\"code_endofline\">;<\/span>          <\/span>\n        <\/td>\n<\/tr>\n<\/table>\n<p>    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(A)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mn>1<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>2<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>3<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>4<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>5<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><br \/>\n    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(B)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mn>1<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>2<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>3<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>4<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>5<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math>\n<\/div>\n<hr \/>\n<p>La primera, A, ser\u00eda una matriz de 2&#215;3, B ser\u00eda una matriz de 3&#215;2. La manera de acceder a los elementos es mediante A[i,j]. <\/p>\n<p>Los operadores anteriores comentados para vectores funcionan igual en matrices. En el caso de elevar una matriz a una potencia tendr\u00edamos que utilizar &quot;<strong>^^<\/strong>&quot;. <\/p>\n<p>Otra forma de acceder a submatrices es con los comandos:<\/p>\n<ul>\n<li><strong>col<\/strong>((<em>Matriz<\/em>,<em>N\u00famColumna<\/em>)): Recupera la columna <em>N\u00famColumna<\/em>. <\/li>\n<li><strong>row<\/strong>((<em>Matriz<\/em>,<em>N\u00famFila<\/em>)): Recupera la fila <em>N\u00famFila<\/em>. <\/li>\n<li><strong>submatrix<\/strong>(\\(i_1,i_2,\\ldots,i_p\\), <em>Matriz<\/em>,\\(j_1,j_2,\\ldots,j_q\\)): Elimina de la <em>Matriz <\/em>las filas cuyos n\u00fameros son \\(i_1,i_2,\\ldots,i_p\\) y las columnas cuyos n\u00fameros son \\(j_1,j_2,\\ldots,j_q\\). No es preciso que est\u00e9n ambas: pueden eliminarse \u00fanicamente filas o columnas.<\/li>\n<li><strong>addrow<\/strong>(<em>Matriz<\/em>, \\(v_1, \\ldots, v_p\\)): A\u00f1ade en la base de <em>Matriz <\/em>las filas dadas por vectores (o matrices) \\(v_1, \\ldots, v_p\\). Las longitudes deben ser concordantes.<\/li>\n<li><strong>addcol<\/strong>(<em>Matriz<\/em>, \\(v_1, \\ldots, v_p\\)): A\u00f1ade en la base de <em>Matriz <\/em>las filas dadas por vectores (o matrices) \\(v_1, \\ldots, v_p\\). Las longitudes deben ser concordantes.<\/li>\n<li><strong>matrix_size<\/strong>(<em>Matriz<\/em>): Proporciona las dimensiones de la matriz.<\/li>\n<li><strong>transpose<\/strong>(<em>Matriz<\/em>): Proporciona la matriz traspuesta de <em>Matriz<\/em>.<\/li>\n<li><strong>mat_trace<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su traza.<\/li>\n<\/ul>\n<p>Algunas matrices interesantes:<\/p>\n<ul>\n<li><strong>diagmatrix<\/strong>(<em>N\u00famero<\/em>,<em>Valor<\/em>): Genera una matriz cuadrada diagonal cuyo tama\u00f1o se establece mediante el valor de <em>N\u00famero <\/em> y en la que todos los elementos de la diagonal tienen el mismo <em>Valor<\/em>. <\/li>\n<li><strong>diag_matrix<\/strong>(\\(a_1,a_2,\\ldots,a_n\\)): Genera una matriz diagonal cuadrada con \\(a_1,a_2,\\ldots,a_n\\) en la diagonal. <\/li>\n<li><strong>ident<\/strong>(<em>N\u00famero<\/em>): Genera la matriz identidad (cuadrada) cuyo tama\u00f1o viene dado por el valor <em>N\u00famero<\/em>; es un caso particular del anterior.<\/li>\n<li><strong>zeromatrix<\/strong>(<em>n<\/em>,<em>m<\/em>): Genera la matriz de <em>n<\/em> filas y <em>m<\/em> columnas en la que todos sus elementos son ceros.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Construir la matriz \\[\\begin{pmatrix}2 &#038; 2 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n2 &#038; 2 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; -1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\end{pmatrix}\\]  <\/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><br \/>\n<button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i7)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">$<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Escribir en forma matricial el sistema \\[\\begin{matrix}x+2y=10\\\\ 2x-2y=4\\\\ 3x+5y=26\\end{matrix}\\]  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i4)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">10<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">26<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{pmatrix}1 &amp; 2\\\\ 2 &amp; -2\\\\ 3 &amp; 5\\end{pmatrix} \\begin{pmatrix}x\\\\ y\\end{pmatrix}=\\begin{pmatrix}10\\\\ 4\\\\ 26\\end{pmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dadas \\(A\\)=[[1,-3],[2,2],[4,1]] y \\(B\\)=[[-1,-7], [4,-5]], Si \\(C=B^t\\cdot A^t\\), \u00bfcu\u00e1nto es \\(c_{12}\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table >\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">    <span class=\"prompt\">(%i4)\t  <\/span>  <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>],[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">7<\/span>], [<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">5<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">C<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span>(<span class=\"code_variable\">A<\/span>.<span class=\"code_variable\">B<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">C<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>]<span class=\"code_endofline\">;<\/span>    <\/span>  <\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(C)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>13<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>8<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>24<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>33<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><br \/>\n    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o4) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mn>6<\/mn>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(A=\\begin{bmatrix} 1 &#038; 3\\\\  \\alpha &#038; 1 \\end{bmatrix}\\), \u00bfcu\u00e1l es el valor de \\(\\alpha\\) para el cual A es una ra\u00edz del polinomio \\(f(x)=x^2-2x-8\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6() {\n  var htmlShow6 = document.getElementById(\"html-show6\");\n  if (htmlShow6.style.display === \"none\") {\n    htmlShow6.style.display = \"block\";\n  } else {\n    htmlShow6.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv6()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i2)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\">  <span class=\"code_variable\">A<\/span>  <span class=\"code_operator\">:<\/span>  <span class=\"code_function\">matrix<\/span>([  <span class=\"code_number\">1<\/span>,  <span class=\"code_number\">3<\/span>], [  <span class=\"code_variable\">x<\/span>,  <span class=\"code_number\">1<\/span>])  <span class=\"code_endofline\">$<\/span>  <br \/>  <span class=\"code_variable\">A<\/span>  <span class=\"code_operator\">^<\/span>  <span class=\"code_operator\">^<\/span>  <span class=\"code_number\">2<\/span>  <span class=\"code_operator\">&#8211;<\/span>  <span class=\"code_number\">2<\/span>  <span class=\"code_operator\">*<\/span>  <span class=\"code_variable\">A<\/span>  <span class=\"code_operator\">&#8211;<\/span>  <span class=\"code_number\">8<\/span>  <span class=\"code_operator\">*<\/span>  <span class=\"code_function\">ident<\/span>(  <span class=\"code_number\">2<\/span>)   <span class=\"code_endofline\">;<\/span><\/span>        <\/td>\n<\/tr>\n<\/table>\n<p>    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o2) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mrow>\n                        <mn>3<\/mn>\n                        <mo>\u2062<\/mo>\n                        <mi>x<\/mi>\n                        <mi>\u2212<\/mi>\n                        <mn>9<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mn>3<\/mn>\n                        <mo>\u2062<\/mo>\n                        <mi>x<\/mi>\n                        <mi>\u2212<\/mi>\n                        <mn>9<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><br \/>\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">%<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">x<\/span>)<span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p>    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o3) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mo>[<\/mo>\n            <mi>x<\/mi>\n            <mi>=<\/mi>\n            <mn>3<\/mn>\n            <mo>]<\/mo>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><\/p>\n<\/div>\n<hr \/>\n<p>A veces, podemos necesitar comandos que nos ayuden a simplificar expresiones; algunos son:<\/p>\n<ul>\n<li><strong>expand<\/strong>(<em>expr<\/em>): Expande la expresi\u00f3n <em>expr<\/em>. Los productos de sumas y de sumas con exponentes se multiplican, los numeradores de las expresiones racionales que son sumas se separan en sus respectivos t\u00e9rminos, y las multiplicaciones (tanto las que son conmutativas como las que no) se distribuyen sobre las sumas en todos los niveles de <em>expr<\/em>.<\/li>\n<li><strong>radcan<\/strong>(<em>expr<\/em>): Simplifica la expresi\u00f3n <em>expr<\/em>, que puede contener logaritmos, exponenciales y radicales, convirti\u00e9ndola a una forma can\u00f3nica, lo que significa que todas las expresiones funcionalmente equivalentes se reducen a una forma \u00fanica.<\/li>\n<li><strong>ratsimp<\/strong>(<em>expr<\/em>): Simplifica la expresi\u00f3n <em>expr <\/em>y todas sus subexpresiones, incluyendo los argumentos de funciones no racionales. <\/li>\n<li><strong>fullratsimp<\/strong>(<em>expr<\/em>): Aplica repetidamente <em>ratsimp<\/em> a una expresi\u00f3n, seguida de simplificaciones no racionales, hasta que no se obtienen m\u00e1s transformaciones; entonces devuelve el resultado. <\/li>\n<\/ul>\n<p>Pod\u00e9is ver m\u00e1s en <a href=\"https:\/\/maxima.sourceforge.io\/docs\/manual\/es\/maxima_47.html\" rel=\"noopener\" target=\"_blank\">Funciones y variables para simplificaci\u00f3n<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Maxima Las pr\u00e1cticas la realizaremos con Maxima, un sistema para la manipulaci\u00f3n de expresiones simb\u00f3licas y num\u00e9ricas. Esta herramienta inform\u00e1tica nos ayudar\u00e1 a resolver problemas de la asignatura de forma sencilla y&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[20],"class_list":["post-96","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\/96","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=96"}],"version-history":[{"count":3,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/96\/revisions"}],"predecessor-version":[{"id":98,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/96\/revisions\/98"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=96"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=96"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}