{"id":473,"date":"2025-11-17T08:32:46","date_gmt":"2025-11-17T07:32:46","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=473"},"modified":"2025-11-20T11:29:23","modified_gmt":"2025-11-20T10:29:23","slug":"alg-el-espacio-vectorial-euclideo","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=473","title":{"rendered":"ALG: El Espacio Vectorial Eucl\u00eddeo"},"content":{"rendered":"<p>Hoy hemos comenzado con el Tema 7. El tema lo hemos llamado Ortogonalizaci\u00f3n, aunque es una parte del m\u00e1s gen\u00e9rico que ser\u00eda Espacio Vectorial Eucl\u00eddeo. El prop\u00f3sito de este tema es dar a un espacio vectorial la herramientas para poder establecer una distancia entre vectores y conseguir encontrar la distancia m\u00ednima entre subespacios o variedades.<\/p>\n<p><strong>Objetivos<\/strong><\/p>\n<ul>\n<li>Conocer y saber determinar un producto escalar y sus propiedades.<\/li>\n<li>Saber calcular la matriz de Gram o m\u00e9trica de un producto escalar<\/li>\n<li>Conocer y saber determinar la norma de un vector y sus propiedades.<\/li>\n<li>Conocer y determinar vectores ortogonales y ortonormales y sus propiedades.<\/li>\n<li>Calcular bases ortonormales.<\/li>\n<li>Conocer el espacio vectorial eucl\u00eddeo can\u00f3nico R<sup>n<\/sup><\/li>\n<li>Conocer y determinar una proyecci\u00f3n ortogonal de un vector.<\/li>\n<li>Saber calcular el complemento ortogonal de un subespacio y sus propiedades.<\/li>\n<li>Conocer y saber calcular transformaciones y matrices ortogonal y sus propiedades<\/li>\n<\/ul>\n<p>Para ello comenzamos con la definici\u00f3n del producto escalar en un espacio vectorial, la norma de un vector, distancia entre dos vectores y el \u00e1ngulo de dos vectores.<\/p>\n<p>Un producto escalar entre dos vectores de un \\(V,\\, \\mathbb{K}\\)-e.v., es una aplicaci\u00f3n \\(\\bullet:V\\times V\\to\\mathbb{K}\\) que verifica:<\/p>\n<ul>\n<li>\\(\\forall\\ u,v\\in V,\\ u\\bullet v=v\\bullet u\\)<\/li>\n<li>\\(\\forall\\ u,v,w\\in V,\\ \\forall\\ \\lambda,\\mu\\in \\mathbb{K},\\ (\\lambda u+\\mu v)\\bullet w=\\lambda(u\\bullet w)+\\mu(v\\bullet w)\\)<\/li>\n<li>\\(\\forall\\ u\\in V,\\ u\\neq 0_V,  u\\bullet u &gt; 0\\)<\/li>\n<li>\\(\\forall\\ u\\in V,\\  u\\bullet u = 0\\Leftrightarrow  u= 0_V\\)<\/li>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(\\mathcal{M}_2(\\mathbb{R})\\) con un producto \\[\\begin{bmatrix}a_{11}&#038;a_{12}\\\\ a_{21}&#038;a_{22}\\end{bmatrix}\\bullet\\begin{bmatrix}b_{11}&#038;b_{12}\\\\ b_{21}&#038;b_{22}\\end{bmatrix}=a_{11}b_{11}+ a_{22}b_{22}\\] Discutir si es un producto escalar.<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d() {\n  var htmlShow41d = document.getElementById(\"html-show41d\");\n  if (htmlShow41d.style.display === \"none\") {\n    htmlShow41d.style.display = \"block\";\n  } else {\n    htmlShow41d.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Producto escalar Ejemplo 1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/k4bEKT-Yk4w?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Estudiar para qu\u00e9 valores de \\(\\alpha\\in\\mathbb{R}\\) la matriz \\(\\begin{bmatrix}2&#038;\\alpha\\\\ 2&#038;3\\end{bmatrix}\\) define un producto escalar en \\(\\mathbb{R}^2.\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d2() {\n  var htmlShow41d2 = document.getElementById(\"html-show41d2\");\n  if (htmlShow41d2.style.display === \"none\") {\n    htmlShow41d2.style.display = \"block\";\n  } else {\n    htmlShow41d2.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Producto Escalar. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/bEjcP86V0cY?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>Un  \\(\\mathbb{K}\\) espacio vectorial dotado con un producto escalar se denomina espacio eucl\u00eddeo, y solemos notarlo como \\((\\mathcal{E},\\bullet)\\). Habitualmente trabajaremos con \\(\\mathbb{R}\\) espacios vectoriales eucl\u00eddeos.\n<\/p>\n<p>Ejemplos de espacio vectorial eucl\u00eddeo con los que trabajaremos:<\/p>\n<ul>\n<li>\\(\\mathbb{R}^n\\), podemos definir el producto escalar eucl\u00eddeo: \\(u\\bullet v=\\sum_{i=1}^nu_iv_i\\) <\/li>\n<li>\\(\\mathcal{M}_n(\\mathbb{R})\\), definimos: \\(A\\bullet B=\\text{tr}(B^tA)\\) <\/li>\n<li>\\(\\mathbb{R}_n[x]\\), definimos: \\(p(x)\\bullet q(x)=\\int_0^1p(x)q(x)\\,dx\\) <\/li>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> Consideremos en  \\([a_{ij}],[b_{ij}]\\mathcal{M}_{m\\times n}(\\mathbb{R})\\), entonces  \\[[a_{ij}]\\bullet[b_{ij}]=\\sum_{i=1}^{m}\\sum_{j=1}^{n}a_{ij}\\cdot b_{ij}\\] es un producto escalar<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d3() {\n  var htmlShow41d3 = document.getElementById(\"html-show41d3\");\n  if (htmlShow41d3.style.display === \"none\") {\n    htmlShow41d3.style.display = \"block\";\n  } else {\n    htmlShow41d3.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d3\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Producto Escalar en un espacio eucl\u00eddeo Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/ynnC0fuAq4k?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Calcular \\(\\begin{bmatrix}1&#038;2&#038;3\\\\ -1&#038;-2&#038;-3\\end{bmatrix}\\bullet\\begin{bmatrix}-4 &#038;-5&#038; -6\\\\ 4&#038;5&#038;6&#038;\\end{bmatrix}\\).<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d4() {\n  var htmlShow41d4 = document.getElementById(\"html-show41d4\");\n  if (htmlShow41d4.style.display === \"none\") {\n    htmlShow41d4.style.display = \"block\";\n  } else {\n    htmlShow41d4.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d4\" style=\"display: none;\">\n\\(\\begin{multline*}\\begin{bmatrix}1&#038;2&#038;3\\\\ -1&#038;-2&#038;-3\\end{bmatrix}\\bullet\\begin{bmatrix}-4 &#038;-5&#038; -6\\\\ 4&#038;5&#038;6\\end{bmatrix}=\\begin{bmatrix}1&#038;2&#038;3\\end{bmatrix}\\bullet\\begin{bmatrix}-4 &#038;-5&#038; -6\\end{bmatrix}+\\\\ \\begin{bmatrix} -1&#038;-2&#038;-3\\end{bmatrix}\\bullet\\begin{bmatrix}4&#038;5&#038;6\\end{bmatrix}=-64\\end{multline*}\\)\n<\/div>\n<hr \/>\n<p>En un \\(\\mathbb{K}\\) espacio vectorial eucl\u00eddeo, \\((\\mathcal{E},\\bullet)\\), podemos definir la norma de un vector  \\(v\\in\\mathcal{E}\\), \\(||v||\\), como \\[||v||=\\sqrt{v\\bullet v}.\\]\n<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Dado el producto escalar habitual en \\(\\mathcal{M}_2(\\mathbb{R})\\), cu\u00e1nto es \\[\\left\\|\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}\\right\\|\\] <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4x() {\n  var htmlShow4x = document.getElementById(\"html-show4x\");\n  if (htmlShow4x.style.display === \"none\") {\n    htmlShow4x.style.display = \"block\";\n  } else {\n    htmlShow4x.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv4x()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4x\" style=\"display: none;\">\nVeamos,<br \/>\n\\[\\begin{split}\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}\\bullet\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}&#038;=tr\\left(\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}^t.\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}\\right)\\\\ &#038;=tr\\left(\\begin{bmatrix}5&#038;1\\\\ 1&#038;2\\end{bmatrix}\\right)\\\\ &#038;=7\\end{split}\\]<br \/>\nLuego, \\(\\left\\|\\begin{bmatrix}-1&#038;1\\\\ 2&#038;1\\end{bmatrix}\\right\\|=\\sqrt{7}\\)\n<\/div>\n<hr \/>\n<blockquote><p><strong>Propiedades:<\/strong> La norma cumple las siguientes propiedades:<\/p>\n<ul>\n<li>\\({\\displaystyle \\|x\\|\\geq 0\\quad \\forall x\\in V}\\). Adem\u00e1s, \\({\\displaystyle \\|x\\|=0\\Leftrightarrow x=0}\\)<\/li>\n<li>\\({\\displaystyle \\|kx\\|=|k|\\cdot \\|x\\|\\quad \\forall x\\in V,\\forall k\\in \\mathbb {K} }\\)<\/li>\n<li>\\({\\displaystyle \\|x+y\\|\\leq \\|x\\|+\\|y\\|\\quad \\forall x,y\\in V}\\)<\/li>\n<\/ul>\n<\/blockquote>\n<hr \/>\n<p>Esta definici\u00f3n, a su vez, nos permite definir la distancia entre dos vectores  \\(u,v\\in\\mathcal{E}\\), \\(\\mathbf{dist}(u,v),\\) como \\[\\mathbf{dist}(u,v)=||u-v||.\\]\n<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Dado el producto escalar habitual en \\(\\mathbb{R}_2[x]\\), cu\u00e1l es \\(\\mathbf{dist}\\left(x^2+2x+1,x+1\\right)\\) <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4x2() {\n  var htmlShow4x2 = document.getElementById(\"html-show4x2\");\n  if (htmlShow4x2.style.display === \"none\") {\n    htmlShow4x2.style.display = \"block\";\n  } else {\n    htmlShow4x2.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv4x2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4x2\" style=\"display: none;\">\nVeamos,<br \/>\n\\[\\begin{split}<br \/>\n\\mathbf{dist}\\left(x^2+2x-1,x+1\\right)&#038;=\\|((x^2+2x+1)-(x+1))\\bullet ((x^2+2x+1)-(x+1))\\| \\\\ &#038;= \\sqrt{\\int_0^1(x^2+x)^2dx}\\\\ &#038;=\\sqrt{\\frac{31}{30}}\\end{split}\\]\n<\/div>\n<hr \/>\n<p>Una \u00faltima, nos permite definir el coseno entre dos vectores  \\(u,v\\in\\mathcal{E}\\), \\(cos(u,v)\\), como \\[cos(u,v)=\\frac{u\\bullet v}{\\|u\\|\\cdot\\|v\\|}.\\]\n<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Dado el producto escalar habitual en \\(\\mathcal{M}_2(\\mathbb{R})\\), cu\u00e1nto es \\[\\cos\\left(\\begin{bmatrix}1&#038;1\\\\ 0&#038;-1\\end{bmatrix},\\begin{bmatrix}1&#038;-1\\\\ 2&#038;2\\end{bmatrix}\\right)\\] <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d5() {\n  var htmlShow41d5 = document.getElementById(\"html-show41d5\");\n  if (htmlShow41d5.style.display === \"none\") {\n    htmlShow41d5.style.display = \"block\";\n  } else {\n    htmlShow41d5.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d5\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Coseno en un Espacio Eucl\u00eddeo. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/a9ZJQ4u1X_o?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<h3>Ortogonalidad<\/h3>\n<p>Esto nos da pie a definir cu\u00e1ndo dos vectores son ortogonales: cuando se de que \\(\\vec{x}\\bullet \\vec{y}=0\\) <\/p>\n<p>As\u00ed pondremos que \\[\\vec{x}\\perp  \\vec{y} \\Leftrightarrow \\vec{x}\\bullet \\vec{y}=0\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean las matrices \\(\\begin{bmatrix}1 &#038; 2\\\\ -1 &#038; 1\\end{bmatrix}\\) y \\(\\begin{bmatrix}0 &#038; b\\\\ 1 &#038; -2\\end{bmatrix}\\), \u00bfcu\u00e1l es el valor de \\(b\\) para que sean ortogonales?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4x28() {\n  var htmlShow4x28 = document.getElementById(\"html-show4x28\");\n  if (htmlShow4x28.style.display === \"none\") {\n    htmlShow4x28.style.display = \"block\";\n  } else {\n    htmlShow4x28.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv4x28()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4x28\" style=\"display: none;\">\nCalculemos el producto escalar:<br \/>\n \\[\\begin{bmatrix}1 &#038; 2\\\\ -1 &#038; 1\\end{bmatrix} \\bullet\\begin{bmatrix}0 &#038; b\\\\ 1 &#038; -2\\end{bmatrix}= \\textbf{tr}\\left(\\begin{bmatrix}0 &#038; b\\\\ 1 &#038; -2\\end{bmatrix}^t\\begin{bmatrix}1 &#038; 2\\\\ -1 &#038; 1\\end{bmatrix}\\right)=<br \/>\n\\textbf{tr}\\left(\\begin{bmatrix}-1 &#038; 1\\\\ b+2 &#038; 2 b-2\\end{bmatrix}\\right)=2 b-3\\]<br \/>\nLuego, para que la el producto escalar sea 0, deber\u00e1 ser \\(b=\\frac{3}{2}\\).\n<\/div>\n<hr \/>\n<p>Con esta definici\u00f3n, decimos que \\(B=\\{\\vec{v}_1,\\vec{v}_2,\\ldots,\\vec{v}_n\\}\\) es un conjunto ortogonal si dos a dos sus vectores son ortogonales; es decir, \\(\\vec{v}_i\\bullet\\vec{v}_j=0\\forall i\\neq j\\)<\/p>\n<h3>Matriz de Gram<\/h3>\n<p>El producto escalar y la norma de un espacio eucl\u00eddeo nos permite definir una m\u00e9trica, esta puede ser expresada mediante la matriz de Gram. Sea \\((E,\\bullet)\\) el espacio vectorial eucl\u00eddeo y \\(B=\\{\\vec{u}_1,\\ldots,\\vec{u}_n\\}\\) una base de \\(E\\), llamamos matriz de Gram, respecto de la base \\(B\\), a la matriz \\(G=[g_{ij}=\\vec{u}_i\\bullet \\vec{u}_j]\\). Notar que la matriz \\(G\\) siempre es sim\u00e9trica. <\/p>\n<p>De este modo, dados \\(\\vec{x}=(x_1,x_2,\\ldots,x_n)\\), \\(\\vec{y}=(y_1,y_2,\\ldots,y_n)\\in E\\), ser\u00e1 \\[\\vec{x}\\bullet \\vec{y}=\\textbf{x}^t\\, G\\, \\textbf{y}=[x_1\\,x_2\\,\\ldots\\,x_n]G\\begin{bmatrix}y_1 \\\\ y_2\\\\ \\vdots \\\\ y_n\\end{bmatrix}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(B:\\{[1,0,-1], [1,2,0], [0,1,-1]\\}\\subset\\mathbb{R}^3\\) una base, en la que se define el producto escalar como \\[\\vec{x}\\bullet\\vec{y}=x_1y_1+3x_2y_2+x_3y_3.\\] \u00bfcu\u00e1nto vale la traza de su matriz de Gram? <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv41d2g() {\n  var htmlShow41d2g = document.getElementById(\"html-show41d2g\");\n  if (htmlShow41d2g.style.display === \"none\") {\n    htmlShow41d2g.style.display = \"block\";\n  } else {\n    htmlShow41d2g.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv41d2g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show41d2g\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matriz de Gram Ejercicio 2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/RjuCTKndnzo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(B:\\{1-X,1-X+X^2,1-X^2\\}\\subset\\mathbb{R}_2[X]\\) una base, en la que se define el producto escalar habitual entre polinomios. \u00bfCu\u00e1nto vale el determinante de su matriz de Gram? <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4d() {\n  var htmlShow4d = document.getElementById(\"html-show4d\");\n  if (htmlShow4d.style.display === \"none\") {\n    htmlShow4d.style.display = \"block\";\n  } else {\n    htmlShow4d.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv4d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4d\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matriz de Gram. Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/ikl9tmeDTJU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>&nbsp;<\/p>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong> Sean \\(S=\\left\\{(x,y,z,t,u)\\in\\mathbb{R}^5; -z-y+3x+t-1=0,\\, 2y+x+u-t=0\\right\\}\\) y \\(T=\\left\\{(x,y,z,t,u)\\in\\mathbb{R}^5;-2z+y-x-u=0,\\, 3z+2t-1=0,\\, -z+y+4x+u-4=0\\right\\}\\) \u00bfcu\u00e1l es su posici\u00f3n relativa?<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>Son paralelas(=)<\/li>\n<li>Son incidentes(-)<\/li>\n<li>Se cortan(+)<\/li>\n<li>Se cruzan(x)<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p>\n<button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>D.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistema04.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hoy hemos comenzado con el Tema 7. El tema lo hemos llamado Ortogonalizaci\u00f3n, aunque es una parte del m\u00e1s gen\u00e9rico que ser\u00eda Espacio Vectorial Eucl\u00eddeo. El prop\u00f3sito de este tema es dar&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-473","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/473","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=473"}],"version-history":[{"count":10,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/473\/revisions"}],"predecessor-version":[{"id":521,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/473\/revisions\/521"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}