{"id":560,"date":"2025-11-24T08:17:22","date_gmt":"2025-11-24T07:17:22","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=560"},"modified":"2025-11-27T11:27:22","modified_gmt":"2025-11-27T10:27:22","slug":"alg-complemento-ortogonal","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=560","title":{"rendered":"ALG: Complemento ortogonal"},"content":{"rendered":"<p>Si tenemos un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, \\(\\mathcal{E}\\), definimos el complemento ortogonal (a veces simplemente ortogonal) de un subespacio \\(S\\) de \\(\\mathcal{E}\\) a \\[S^\\bot=\\{\\vec{v}\\in \\mathcal{E}|\\;\\vec{v}\\bullet\\vec{u}=0\\,\\forall \\vec{u}\\in S\\}\\]<\/p>\n<blockquote><p><strong>Proposici\u00f3n<\/strong>. Si \\(S\\subset E\\) es un subespacio de un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, entonces \\(S^\\bot\\) es un subespacio vectorial.<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(A=\\left\\{\\begin{bmatrix}a&#038;b\\\\ c&#038; 0\\end{bmatrix}\\in\\mathcal{M}_2(\\mathbb{R})\\right\\}\\), \u00bfcu\u00e1l es su complemento ortogonal?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13a() {\n  var htmlShow13a = document.getElementById(\"html-show13a\");\n  if (htmlShow13a.style.display === \"none\") {\n    htmlShow13a.style.display = \"block\";\n  } else {\n    htmlShow13a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13a\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento ortogonal. Ejercicio 1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/CRmzUYVk4oQ?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> Hallar el complemento ortogonal de la variedad lineal de \\(\\mathbb{R}^4\\), que resulta de la intersecci\u00f3n de los hiperplanos \\(x+5y-2z=0\\), y, \\(x+y-z+u=0\\). <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13() {\n  var htmlShow13 = document.getElementById(\"html-show13\");\n  if (htmlShow13.style.display === \"none\") {\n    htmlShow13.style.display = \"block\";\n  } else {\n    htmlShow13.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento ortogonal Ej. 2- Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/E5wVjikL4Lc?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 \\(A=\\left\\{\\begin{bmatrix}2a+b&#038;b\\\\ -b&#038; a-b\\end{bmatrix}\\in\\mathcal{M}_2(\\mathbb{R})\\right\\}\\), \u00bfcu\u00e1l es su complemento ortogonal?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13b() {\n  var htmlShow13b = document.getElementById(\"html-show13b\");\n  if (htmlShow13b.style.display === \"none\") {\n    htmlShow13b.style.display = \"block\";\n  } else {\n    htmlShow13b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13b\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento Ortogonal Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/uLqFhdvU8VA?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 \\(S=\\mathbf{Gen}\\{X^2-2\\}\\subset (\\mathbb{R}_2[X],\\bullet)\\), \u00bfcu\u00e1l es una base de su ortogonal?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1() {\n  var htmlShow1 = document.getElementById(\"html-show1\");\n  if (htmlShow1.style.display === \"none\") {\n    htmlShow1.style.display = \"block\";\n  } else {\n    htmlShow1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Sabemos que el ortogonal de un subespacio generado por un vector, \\(S=\\mathbf{Gen}\\{\\vec{v}\\}\\subset (\\mathcal{E},\\bullet)\\), es el que tiene por ecuaci\u00f3n impl\u00edcita el producto escalar de un vector gen\u00e9rico por dicho vector: \\[S^\\perp=\\{\\vec{u}\\in \\mathcal{E}|\\;\\vec{v}\\bullet\\vec{u}=0\\,\\forall \\vec{u}\\in S\\} \\]<\/p>\n<p>En nuestro caso deber\u00e1 verificarse:<\/p>\n<\/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\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>,<span class=\"code_variable\">c<\/span>].[<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">integrate<\/span>(<span class=\"code_variable\">p<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">q<\/span>,<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">1<\/span>)<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/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>(pq)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mfrac><mrow><mn>28<\/mn><mo>\u2062<\/mo><mi>c<\/mi><mo>+<\/mo><mn>45<\/mn><mo>\u2062<\/mo><mi>b<\/mi><mo>+<\/mo><mn>100<\/mn><mo>\u2062<\/mo><mi>a<\/mi><\/mrow><mn>60<\/mn><\/mfrac><mi>=<\/mi><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Resolvamos el sistema formado por esta sola ecuaci\u00f3n:<\/p>\n<\/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\">sol<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span>(<span class=\"code_variable\">pq<\/span>,[<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>,<span class=\"code_variable\">c<\/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>(sol)<\/mtext><\/mtd><mtd><mo>[<\/mo><mi>a<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mfrac><mrow><mn>45<\/mn><mo>\u2062<\/mo><mi>%r2<\/mi><mo>+<\/mo><mn>28<\/mn><mo>\u2062<\/mo><mi>%r1<\/mi><\/mrow><mn>100<\/mn><\/mfrac><mo>,<\/mo><mi>b<\/mi><mi>=<\/mi><mi>%r2<\/mi><mo>,<\/mo><mi>c<\/mi><mi>=<\/mi><mi>%r1<\/mi><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Por tanto, asignando valores a los par\u00e1metros, obtendremos los dos vectores l.i. de la base del ortogonal:<\/p>\n<\/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_variable\">id<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_function\">length<\/span>(<span class=\"code_variable\">%rnum_list<\/span>))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">vt<\/span><span class=\"code_operator\">:<\/span>[]<span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_function\">length<\/span>(<span class=\"code_variable\">%rnum_list<\/span>) do(<br \/> <span class=\"code_variable\">l1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span>(<span class=\"code_variable\">%rnum_list<\/span>[<span class=\"code_variable\">j<\/span>]<span class=\"code_operator\">=<\/span><span class=\"code_variable\">id<\/span>[<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">j<\/span>],<span class=\"code_variable\">j<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>),<br \/><span class=\"code_variable\">vt<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span>(<span class=\"code_variable\">vt<\/span>,[<span class=\"code_function\">ev<\/span>([<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>,<span class=\"code_variable\">c<\/span>],<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">sol<\/span>,<span class=\"code_variable\">l1<\/span>))]),<br \/> \u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_variable\">vt<\/span>[<span class=\"code_variable\">i<\/span>])<br \/>)<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\/><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mfrac><mn>7<\/mn><mn>25<\/mn><\/mfrac><mo>,<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo>]<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mfrac><mn>9<\/mn><mn>20<\/mn><\/mfrac><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn><mo>]<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea la variedad lineal de \\(\\mathbb{R}^4\\), que resulta de la intersecci\u00f3n de los hiperplanos \\(2x+y-z=0\\), y, \\(x-y+3t=0\\), y \\(u:[-1,a,-1,-9]\\). \u00bfcu\u00e1l es el valor de \\(a\\) para que \\(u\\) perteneca al ortogonal de la variedad lineal?. <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13c() {\n  var htmlShow13c = document.getElementById(\"html-show13c\");\n  if (htmlShow13c.style.display === \"none\") {\n    htmlShow13c.style.display = \"block\";\n  } else {\n    htmlShow13c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13c\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento ortogonal. Ej.7 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/HT6NB5Rli1M?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 \\(\\pi=\\mathbf{Gen}\\left\\{\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1 \\end{bmatrix}, \\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3 \\end{bmatrix}\\right\\}\\). Dados \\(a\\) y \\(b\\) con \\(\\begin{bmatrix}a&#038;1\\\\ 4&#038; b\\end{bmatrix}\\in\\pi^\\bot\\). \u00bfCu\u00e1nto es \\(\\begin{bmatrix}a&#038;1\\\\ 4&#038; b\\end{bmatrix}\\bullet\\begin{bmatrix}2&#038;5\\\\ 1&#038; -3\\end{bmatrix}\\)? <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13d() {\n  var htmlShow13d = document.getElementById(\"html-show13d\");\n  if (htmlShow13d.style.display === \"none\") {\n    htmlShow13d.style.display = \"block\";\n  } else {\n    htmlShow13d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13d\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento ortogonal. Ej.6 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/qZMdECIVLI4?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 \\(S=\\mathbf{Gen}\\left \\{ X^2-2,\\ X-X^2 \\right \\}\\in\\mathbb{R}_2[X]\\),\u00bfcu\u00e1l es el producto escalar de \\((X-1)\\) por el vector unitario del complemento ortogonal de S?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1f() {\n  var htmlShow1f = document.getElementById(\"html-show1f\");\n  if (htmlShow1f.style.display === \"none\") {\n    htmlShow1f.style.display = \"block\";\n  } else {\n    htmlShow1f.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1f()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1f\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Complemento ortogonal. Ej.4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/q-5cLmcUO5A?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<h2>Propiedades<\/h2>\n<blockquote>\n<p><strong>Proposici\u00f3n<\/strong>. Si \\(S,T\\subset \\mathcal{E}\\) son subespacios de un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, y \\(T\\subset S\\)entonces \\(S^\\bot \\subset T^\\bot\\).\n<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Proposici\u00f3n<\/strong>. Si \\(S,T\\subset \\mathcal{E}\\), son subespacios de un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, entonces<\/p>\n<ul>\n<li> \\((S+T)^\\bot=S^\\bot \\cap T^\\bot\\)<\/li>\n<li> \\((S\\cap T)^\\bot=S^\\bot + T^\\bot\\)<\/li>\n<\/ul>\n<\/blockquote>\n<blockquote>\n<p><strong>Proposici\u00f3n<\/strong>. Si \\(S\\subset \\mathcal{E}\\) es un subespacio de un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, entonces $$\\mathcal{E}=S\\oplus S^{\\bot}$$\n<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Corolario<\/strong>. Si \\(S\\subset \\mathcal{E}\\) es un subespacio de un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, entonces $$dim(\\mathcal{E})=dim(S)+ dim(S^{\\bot})$$\n<\/p>\n<\/blockquote>\n<p>Esta \u00faltima Proposici\u00f3n nos dice que \\(\\mathcal{E}\\) es suma directa de \\(S\\) y \\(S^{\\bot}\\); es decir, para todo \\(\\vec{v}\\in \\mathcal{E}\\) existen dos \u00fanicos vectores \\(\\vec{s}_1\\in S\\) y \\(\\vec{s}_2\\in S^{\\bot}\\), tales que $$\\vec{v}=\\vec{s}_1+\\vec{s}_2.$$<\/p>\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>Sea B={(2,1,1),(1,0,10),(2,-3,11)} una base de \\(\\mathbb{R}^3\\), \u00bfcu\u00e1l es el la suma de las normas al cuadrado de una base ortogonal obtenida por un proceso de ortogonalizaci\u00f3n de Gram\u2013Schmidt? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>647\/7<\/li>\n<li>42\/11<\/li>\n<li>562\/9<\/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><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>A.)<\/strong><\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Sea B={(2,1,1),(1,0,10),(2,-3,11)} una base de \\(\\mathbb{R}^3\\), \u00bfcu\u00e1l es el la suma de las normas al cuadrado de una base ortogonal obtenida por un proceso de ortogonalizaci\u00f3n de Gram\u2013Schmidt?<\/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=\"input\"><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">10<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">11<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v1<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ 2\\operatorname{,}1\\operatorname{,}1\\right] \\]<\/p>\n<p>\\[\\left[ -3\\operatorname{,}-2\\operatorname{,}8\\right] \\]<\/p>\n<p>\\[\\left[ \\frac{10}{7}\\operatorname{,}-\\frac{19}{7}\\operatorname{,}-\\frac{1}{7}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La respuesta 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\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">u1<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u1<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u2<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">u3<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">u3<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{647}{7}\\]<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Si tenemos un espacio vectorial eucl\u00eddeo de dimensi\u00f3n finita, \\(\\mathcal{E}\\), definimos el complemento ortogonal (a veces simplemente ortogonal) de un subespacio \\(S\\) de \\(\\mathcal{E}\\) a \\[S^\\bot=\\{\\vec{v}\\in \\mathcal{E}|\\;\\vec{v}\\bullet\\vec{u}=0\\,\\forall \\vec{u}\\in S\\}\\] Proposici\u00f3n. Si \\(S\\subset&#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-560","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/560","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=560"}],"version-history":[{"count":3,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/560\/revisions"}],"predecessor-version":[{"id":616,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/560\/revisions\/616"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=560"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=560"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=560"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}