{"id":468,"date":"2026-11-11T08:15:09","date_gmt":"2026-11-11T07:15:09","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=468"},"modified":"2026-10-29T18:30:23","modified_gmt":"2026-10-29T17:30:23","slug":"alg-el-espacio-vectorial-euclideo","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=468","title":{"rendered":"ALG: El Espacio Vectorial Eucl\u00eddeo"},"content":{"rendered":"<p>Comenzamos 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 las 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 ortogonales 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]\\), \u00bfcu\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 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 permiten 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(id) {\n  var htmlShow = document.getElementById(id);\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<p><button onclick=\"showHtmlDiv('html-show-1')\">Soluci\u00f3n:<\/button><\/p><div id=\"html-show-1\" style=\"display: none;\"><\/p>\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><br \/>\n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Comenzamos 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&#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":[],"class_list":["post-468","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/468","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=468"}],"version-history":[{"count":3,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/468\/revisions"}],"predecessor-version":[{"id":471,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/468\/revisions\/471"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=468"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=468"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}