{"id":358,"date":"2026-10-19T15:14:42","date_gmt":"2026-10-19T13:14:42","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=358"},"modified":"2026-10-17T20:25:24","modified_gmt":"2026-10-17T18:25:24","slug":"mathbio-minimos-cuadrados-autovalores-autovectores-y-funciones-reales","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=358","title":{"rendered":"MathBio: Sistemas, m\u00ednimos cuadrados, autovalores, autovectores y funciones reales"},"content":{"rendered":"<h2>Sistemas de ecuaciones<\/h2>\n<p>En pasados d\u00edas vimos c\u00f3mo resolv\u00edamos sistemas de ecuaciones y, en particular, el problema de m\u00ednimos cuadrados mediante matrices. Hoy abordaremos estos problemas utilizando maxima.<\/p>\n<ul>\n<li><strong>linsolve<\/strong>(\\([eq_1, &#8230;, eq_m], [x_1, &#8230;, x_n]\\)): Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}2x+y-z=1 \\\\x-3y+2z=1 \\\\ -x+2y-4z=2\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1a() {\n  var htmlShow1a = document.getElementById(\"html-show1a\");\n  if (htmlShow1a.style.display === \"none\") {\n    htmlShow1a.style.display = \"block\";\n  } else {\n    htmlShow1a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_sist01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>El sistema puede tener infinitas soluciones, en cuyo caso estas se dan en forma param\u00e9trica:<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}3x-y-z+t=2 \\\\ x+y-2z-5t=1 \\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a() {\n  var htmlShow2a = document.getElementById(\"html-show2a\");\n  if (htmlShow2a.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a\" 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\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">eq1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">eq2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">sol<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">eq1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ x=\\frac{3 {\\mathrm{\\% r2}}+4 {\\mathrm{\\% r1}}+3}{4}\\operatorname{,}y=\\frac{5 {\\mathrm{\\% r2}}+16 {\\mathrm{\\% r1}}+1}{4}\\operatorname{,}z={\\mathrm{\\% r2}}\\operatorname{,}t={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El punto vendr\u00e1 dado con los par\u00e1metros %r1=0 y %r2=0<\/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\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora, separamos los par\u00e1metros %r1 y %r2. Para %r1, tenemos los coeficientes que multiplican a %r1:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">16<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para %r2, tenemos los coeficientes que multiplican a %r2:<\/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\">(%i6)<\/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\">3<\/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_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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\">As\u00ed pues, los vectores directores ser\u00e1n v y u, y las ecuaciones param\u00e9tricas son<\/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\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">print<\/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_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/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_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;+ \u03bb&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <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_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;+ \u03bc&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <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_variable\">u<\/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>\\[\\begin{bmatrix}x\\\\y\\\\z\\\\t\\end{bmatrix}=\\begin{bmatrix}\\frac{3}{4}\\\\\\frac{1}{4}\\\\0\\\\0\\end{bmatrix}+ \\lambda \\begin{bmatrix}1\\\\4\\\\0\\\\1\\end{bmatrix}+ \\mu \\begin{bmatrix}\\frac{3}{4}\\\\\\frac{5}{4}\\\\1\\\\0\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<h2>M\u00ednimos cuadrados<\/h2>\n<p>La determinaci\u00f3n del cero absoluto proviene del trabajo del f\u00edsico franc\u00e9s Jacques Charles (1746-1823), quien descubri\u00f3 que <strong>el volumen de un gas a presi\u00f3n constante crece de manera lineal con respecto a la temperatura<\/strong>.<\/p>\n<p>\\[\\begin{array}{l|cccc}<br \/>\nT &#038; -40 &#038; -20 &#038; 0 &#038;  \\\\ \\hline<br \/>\nV &#038; 19.1482 &#038; 20.7908 &#038;  22.4334  \\\\ \\hline<br \/>\nT &#038; 20 &#038; 40&#038; 60&#038; 80 \\\\ \\hline<br \/>\nV &#038;  24.0760 &#038;  25.7186 &#038; 27.3612 &#038; 29.0038 \\\\ \\hline<br \/>\n\\end{array}\\]<\/p>\n<p>En la tabla se observa la relaci\u00f3n entre volumen y temperatura. Para crear los valores que aparecen en la tabla, una mol de hidr\u00f3geno se mantiene a una presi\u00f3n constante de una atm\u00f3sfera. El volumen V es aproximado y se mide en litros, y la temperatura T se mide en grados Celsius.<\/p>\n<p>Empleando dichos puntos, se puede determinar que \\(T\\) y \\(V\\) se relacionan a trav\u00e9s de la ecuaci\u00f3n lineal. Veamos c\u00f3mo lo hizo.<\/p>\n<p>Tratamos de buscar una relaci\u00f3n \\(V=aT+b\\) y para ello consideramos el sistema:<br \/>\n\\[\\begin{align*}<br \/>\n-40a+b&#038;= 19.1482 \\\\<br \/>\n-20a+b&#038;= 20.7908\\\\<br \/>\n0a+b&#038;= 22.4334\\\\<br \/>\n20a+b&#038;= 24.0760\\\\<br \/>\n40a+b&#038;= 25.7186\\\\<br \/>\n60a+b&#038;= 27.3612\\\\<br \/>\n80a+b&#038;= 29.0038<br \/>\n\\end{align*}\\]<\/p>\n<p>Es f\u00e1cil observar que el sistema no tiene soluci\u00f3n, pero podemos encontrar la soluci\u00f3n por m\u00ednimos cuadrados, que ser\u00eda la m\u00e1s cercana a una posible soluci\u00f3n.<\/p>\n<p>Te toca: \u00bfc\u00f3mo la calcular\u00edas?<br \/>\n<script>\nfunction showHtmlDiv2d5() {\n  var htmlShow2d5 = document.getElementById(\"html-show2d5\");\n  if (htmlShow2d5.style.display === \"none\") {\n    htmlShow2d5.style.display = \"block\";\n  } else {\n    htmlShow2d5.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2d5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d5\" style=\"display: none;\">\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_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_operator\">\u2212<\/span><span class=\"code_number\">40<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">40<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">60<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">80<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">19<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">1482<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">7908<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">4334<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_number\">24<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">0760<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">25<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">7186<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">27<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">3612<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">29<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">0038<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos las matrices del sistema. Como la matriz A tiene rango dos, planteamos \\(A^tA\\)<\/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\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">AtA<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}14000 &amp; 140\\\\140 &amp; 7\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esta es una matriz regular:<\/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\">invAtA<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">AtA<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{11200} &amp; -\\frac{1}{560}\\\\-\\frac{1}{560} &amp; \\frac{5}{28}\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La soluci\u00f3n que buscamos vendr\u00e1 dada mediante<\/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\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">xs<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">invAtA<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0.08213000000000009\\\\22.4334\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">As\u00ed pues, la ecuaci\u00f3n que buscamos es<\/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\">(%i6) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/span> \u00a0 <span class=\"code_comment\">\/* Total number of significant digits to print *\/<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">&quot;V=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">xs<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;T+&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">xs<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\]\\[\\mbox{}\\\\V=0.08213T+22.4334\\]<\/p>\n<\/div>\n<hr \/>\n<p>Una vez calculada la funci\u00f3n lineal, solo resta saber cu\u00e1nto nos equivocamos con nuestra soluci\u00f3n aproximada por m\u00ednimos cuadrados. De nuevo, te toca resolverlo.<\/p>\n<h2>Autovectores y autovalores<\/h2>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}1 &#038; 1 &#038; 0\\\\<br \/>\n2 &#038; 0 &#038; 2\\\\ -3 &#038; -2 &#038; -1\\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 <!-- 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_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">2<\/span>],[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_variable\">A<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_number\">3<\/span>)))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">ecu_caract<\/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\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup>  <mi>x<\/mi>  <mn>3<\/mn><\/msup><mi>\u2212<\/mi><mi>x<\/mi><mi>=<\/mi><mn>0<\/mn> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o3) <\/mtext> <\/mtd> <mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mi>%i<\/mi><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mi>%i<\/mi><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>0<\/mn><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}4 &#038; 3 &#038; 0\\\\<br \/>\n-6 &#038; -5 &#038; 0\\\\ 3 &#038; 3 &#038; 1\\end{bmatrix}\\]\n<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a8() {\n  var htmlShow2a8 = document.getElementById(\"html-show2a8\");\n  if (htmlShow2a8.style.display === \"none\") {\n    htmlShow2a8.style.display = \"block\";\n  } else {\n    htmlShow2a8.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a8()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a8\" 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\">(%i3) <\/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\">4<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">6<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">5<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_variable\">A<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_number\">3<\/span>)))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">ecu_caract<\/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\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup>  <mi>x<\/mi>  <mn>3<\/mn><\/msup><mo>+<\/mo><mn>3<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mi>=<\/mi><mn>0<\/mn> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o3) <\/mtext> <\/mtd> <mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>1<\/mn><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovectores de la matriz \\[\\begin{bmatrix}4 &#038; 0 &#038; -3\\\\<br \/>\n3 &#038; -1 &#038; 3\\\\<br \/>\n6 &#038; 0 &#038; 5\\end{bmatrix}\\] correspondientes a su autovalor real.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2b() {\n  var htmlShow2b = document.getElementById(\"html-show2b\");\n  if (htmlShow2b.style.display === \"none\") {\n    htmlShow2b.style.display = \"block\";\n  } else {\n    htmlShow2b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b\" 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\">(%i3) <\/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_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/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_operator\">[<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">ecu_caract<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-{{x}^{3}}+8 {{x}^{2}}-29 x-38=0\\]<\/p>\n<p>\\[{ }\\left[ x=-\\frac{\\sqrt{71} \\% i-9}{2}{,}x=\\frac{\\sqrt{71} \\% i+9}{2}{,}x=-1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Como solo hay un autovalor real:<\/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\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_endofline\">.<\/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_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}5 x-3 z\\\\3 z+3 x\\\\6 z+6 x\\end{bmatrix}\\]<\/p>\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_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[solve: dependent equations eliminated: (3)\\]<\/p>\n<p>\\[{ }\\left[ x=0{,}y={\\mathrm{\\% r1}}{,}z=0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esto nos proporciona el vector<\/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\">(%i6) <\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ 0{,}1{,}0\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dados los autovectores de la matriz \\[\\begin{bmatrix}-4 &#038; 0 &#038; -3\\\\<br \/>\n3 &#038; -1 &#038; 3\\\\<br \/>\n6 &#038; 0 &#038; 5\\end{bmatrix}\\] del mayor autovalor real, \u00bfcu\u00e1l es el producto escalar de [1,1,1] por uno de dichos vectores unitarios?.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2c() {\n  var htmlShow2c = document.getElementById(\"html-show2c\");\n  if (htmlShow2c.style.display === \"none\") {\n    htmlShow2c.style.display = \"block\";\n  } else {\n    htmlShow2c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2c\" 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\">(%i3) <\/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_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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/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_operator\">[<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">ecu_caract<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-{{x}^{3}}+3 x+2=0\\]<\/p>\n<p>\\[{ }\\left[ x=2{,}x=-1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El autovalor real mayor es 2<\/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\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_endofline\">.<\/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_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{pmatrix}-3 z-6 x\\\\3 z-3 y+3 x\\\\3 z+6 x\\end{pmatrix}\\]<\/p>\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_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\]\\[solve: dependent equations eliminated: (3)\\]<\/p>\n<p>\\[{ }\\left[ x=-\\frac{{\\mathrm{\\% r1}}}{2}{,}y=\\frac{{\\mathrm{\\% r1}}}{2}{,}z={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esto nos proporciona el vector<\/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\">(%i6) <\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ -\\frac{1}{2}{,}\\frac{1}{2}{,}1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora solo tenemos que calcular<\/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\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/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_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/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>\\[{ }0.8164965809277262\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(v\\) el autovector de mayor norma de los autovectores de los autovalores reales de la matriz \\[\\begin{bmatrix}1 &#038; 1 &#038; 0\\\\ 2 &#038; 0 &#038; 2\\\\ -3 &#038; -2 &#038; -1\\end{bmatrix}.\\] \u00bfCu\u00e1l es el valor de \\([1,5,6].v\/\\| v\\|\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2c4() {\n  var htmlShow2c4 = document.getElementById(\"html-show2c4\");\n  if (htmlShow2c4.style.display === \"none\") {\n    htmlShow2c4.style.display = \"block\";\n  } else {\n    htmlShow2c4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2c4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2c4\" 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\">(%i3) <\/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_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/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_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/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\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">ecu_caract<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-{{x}^{3}}-x=0\\]<\/p>\n<p>\\[{ }\\left[ x=-\\% i{,}x=\\% i{,}x=0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Como solo hay un autovalor real y dos complejos, solo habr\u00e1 un autovector para el autovalor real:<\/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\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/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\">)<\/span><span class=\"code_endofline\">.<\/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_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/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\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/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_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}y+x\\\\2 z+2 x\\\\-z-2 y-3 x\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (3)\\]<\/p>\n<p>\\[{ }\\left[ x=-{\\mathrm{\\% r1}}{,}y={\\mathrm{\\% r1}}{,}z={\\mathrm{\\% r1}}\\right] \\]<\/p>\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\">(%i7) <\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/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_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/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>\\[ 5.773502 \\]<\/p>\n<\/div>\n<hr \/>\n<h2>Funciones<\/h2>\n<p>Una funci\u00f3n ordinaria es aquella que ha sido construida mediante cualquiera de los m\u00e9todos <strong>define<\/strong> o <strong>:=<\/strong> y que es invocada utilizando par\u00e9ntesis.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determina la suma de los valores de la funci\u00f3n \\(f(x)=x\\,e^{1-x^2}\\) para \\(x_i=\\frac{i}{4}\\) para \\(i\\in\\{1,\\ldots,5\\}\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1s() {\n  var htmlShow1s = document.getElementById(\"html-show1s\");\n  if (htmlShow1s.style.display === \"none\") {\n    htmlShow1s.style.display = \"block\";\n  } else {\n    htmlShow1s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1s\" 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\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_number\">1<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">val<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">i<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span>),<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">sum<\/span>(<span class=\"code_variable\">val<\/span>[<span class=\"code_variable\">i<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>),<span class=\"code_variable\">numer<\/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\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(val)<\/mtext> <\/mtd> <mtd><mo>[<\/mo><mfrac><msup> <mi>%e<\/mi> <mfrac><mn>15<\/mn><mn>16<\/mn> <\/mfrac><\/msup><mn>4<\/mn><\/mfrac><mo>,<\/mo><mfrac><msup> <mi>%e<\/mi> <mfrac><mn>3<\/mn><mn>4<\/mn> <\/mfrac><\/msup><mn>2<\/mn><\/mfrac><mo>,<\/mo><mfrac><mrow> <mn>3<\/mn> <mo>\u2062<\/mo> <msup><mi>%e<\/mi><mfrac><mn>7<\/mn><mn>16<\/mn><\/mfrac> <\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><mfrac><mrow> <mn>5<\/mn> <mo>\u2062<\/mo> <msup><mi>%e<\/mi><mrow><mi>\u2212<\/mi><mfrac> <mn>9<\/mn> <mn>16<\/mn><\/mfrac><\/mrow> <\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o3) <\/mtext> <\/mtd> <mtd><mn>4.570748627711323<\/mn> <\/mtd><\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<ul>\n<li><strong>define<\/strong>(\\(f(x_1,\\ldots, x_n)\\), <em>expr<\/em>): Define una funci\u00f3n de nombre \\(f\\) con argumentos \\(x_1,\\ldots, x_n\\) y cuerpo <em>expr<\/em>. <strong>define<\/strong> eval\u00faa siempre su segundo argumento, a menos que se indique lo contrario con el operador de comilla simple. <\/li>\n<\/ul>\n<p>Esta funci\u00f3n la utilizaremos m\u00e1s adelante.<\/p>\n<h2>Gr\u00e1fica de una funci\u00f3n<\/h2>\n<ul>\n<li><strong>wxplot2d<\/strong>(expresi\u00f3n, [variable,m\u00ednimo,m\u00e1ximo],opciones): La funci\u00f3n <strong>wxplot2d<\/strong> representa uno o m\u00e1s gr\u00e1ficos en dos dimensiones. Las expresiones o nombres de funciones que se utilicen para definir curvas deben depender todas ellas de una \u00fanica variable var, siendo obligatorio utilizar x_range para nombrar la variable y darle sus valores m\u00ednimo y m\u00e1ximo usando la siguiente sintaxis: [variable, min, max]. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Dibuja la funci\u00f3n \\(f(x)=x\\,e^{1-x^2}\\) para \\(x\\in[-3,3]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1as() {\n  var htmlShow1as = document.getElementById(\"html-show1as\");\n  if (htmlShow1as.style.display === \"none\") {\n    htmlShow1as.style.display = \"block\";\n  } else {\n    htmlShow1as.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1as()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1as\" 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\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_number\">1<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">wxplot2d<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),[<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">3<\/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\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%t2) <\/mtext><\/mtd><mtd\/><\/mlabeledtr><\/mtable><\/math><img decoding=\"async\" src=\"https:\/\/uploads.jesussoto.es\/doc\/Ejer_concavo01_1.png\" width=\"598\" style=\"max-width:90%;\" loading=\"lazy\" alt=\" (Gr\u00e1ficos) \"\/><br \/><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd\/><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>Sistemas de ecuaciones En pasados d\u00edas vimos c\u00f3mo resolv\u00edamos sistemas de ecuaciones y, en particular, el problema de m\u00ednimos cuadrados mediante matrices. Hoy abordaremos estos problemas utilizando maxima. linsolve(\\([eq_1, &#8230;, eq_m], [x_1,&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[21],"class_list":["post-358","post","type-post","status-publish","format-standard","hentry","category-mathbio","tag-practicas-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/358","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=358"}],"version-history":[{"count":7,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/358\/revisions"}],"predecessor-version":[{"id":372,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/358\/revisions\/372"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}