{"id":730,"date":"2025-12-17T08:20:37","date_gmt":"2025-12-17T07:20:37","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=730"},"modified":"2025-12-16T11:00:07","modified_gmt":"2025-12-16T10:00:07","slug":"alg-ejercicios-de-repaso","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=730","title":{"rendered":"ALG: Ejercicios de repaso"},"content":{"rendered":"<blockquote><p><strong>Ejercicio:<\/strong> Sea la matriz \\(A=\\begin{bmatrix}4 &#038; 2 &#038; 0 &#038; 0\\\\ 3 &#038; 3 &#038; 0 &#038; 0\\\\ 0 &#038; 0 &#038; 2 &#038; 5\\\\ 0 &#038; 0 &#038; 0 &#038; 2\\end{bmatrix}\\) y <strong>v<\/strong> el autovector unitario asociado a su mayor autovalor. \u00bfCu\u00e1l es la distancia de <strong>v<\/strong> a [1,3,6,9]?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv21b() {\n  var htmlShow21b = document.getElementById(\"html-show21b\");\n  if (htmlShow21b.style.display === \"none\") {\n    htmlShow21b.style.display = \"block\";\n  } else {\n    htmlShow21b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv21b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show21b\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Autovectores. Ej.4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/XGt3l0QmmGA?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>\n<p><strong>Ejercicio:<\/strong> Sea la matriz \\(A=\\begin{bmatrix}4 &#038; 1 &#038; 0 &#038; 1\\\\ 2 &#038; 3 &#038; 0 &#038; 1\\\\ -2 &#038; 1 &#038; 2 &#038; -3\\\\ 2 &#038; -1 &#038; 0 &#038; 5\\end{bmatrix}.\\) Cu\u00e1l de los siguientes vectores pertenece al ortogonal del subespacio propio del autovalor \\(\\lambda\\), tal que \\(m_a(\\lambda)=2\\): <\/p>\n<div align='center'>A.)[2,1,0,1]   B.)[3,0,2,1]   C.)[-1,1,2,0]<\/div>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv21c() {\n  var htmlShow21c = document.getElementById(\"html-show21c\");\n  if (htmlShow21c.style.display === \"none\") {\n    htmlShow21c.style.display = \"block\";\n  } else {\n    htmlShow21c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv21c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show21c\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Autovectores. Ej.5 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/DZqTzafDrk8?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>Ejercicio:<\/strong> Sea la matriz \\[A=\\begin{bmatrix}4 &#038; 1 &#038; 0 &#038; 1\\\\ 2 &#038; 3 &#038; 0 &#038; 1\\\\ -2 &#038; 1 &#038; 2 &#038; -3\\\\<br \/>\n2 &#038; -1 &#038; 0 &#038; 5\\end{bmatrix}.\\] Cu\u00e1l es la norma de la proyecci\u00f3n de [1,3,6,9] sobre el ortogonal del subespacio propio del menor autovalor \\(\\lambda\\), tal que \\(m_a(\\lambda)=1\\). <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv21() {\n  var htmlShow21 = document.getElementById(\"html-show21\");\n  if (htmlShow21.style.display === \"none\") {\n    htmlShow21.style.display = \"block\";\n  } else {\n    htmlShow21.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv21()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show21\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Autovectores. Ej.6 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/MKZlJXs67yE?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>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}4 &#038; 1 &#038; 0 &#038; 1\\\\ 2 &#038; 3 &#038; 0 &#038; 1\\\\ -2 &#038; 1 &#038; 2 &#038; -3\\\\<br \/>\n2 &#038; -1 &#038; 0 &#038; 5\\end{bmatrix},\\] \u00bfcu\u00e1l es la norma de su polinomio caracter\u00edstico?. <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv21t() {\n  var htmlShow21t = document.getElementById(\"html-show21t\");\n  if (htmlShow21t.style.display === \"none\") {\n    htmlShow21t.style.display = \"block\";\n  } else {\n    htmlShow21t.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv21t()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show21t\" 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_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\">1<\/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 class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">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_operator\">[<\/span><span class=\"code_operator\">\u2212<\/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\">2<\/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\">2<\/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\">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\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/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\">4<\/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}4 &amp; 1 &amp; 0 &amp; 1\\\\2 &amp; 3 &amp; 0 &amp; 1\\\\-2 &amp; 1 &amp; 2 &amp; -3\\\\2 &amp; -1 &amp; 0 &amp; 5\\end{bmatrix}\\]<\/p>\n<p>\\[{{x}^{4}}-14 {{x}^{3}}+68 {{x}^{2}}-136 x+96\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Una vez obtenido el polinomio caracter\u00edstico, determinamos su norma. Recordemos que el polinomio caracter\u00edstico pertenece al espacio vectorial \\(\\mathbb{R}_4[x]\\), y en \u00e9l definimos el producto escalar habitual como \\[\\int_0^1p(x)q(x)\\ dx\\] para \\(p(x),q(x)\\in\\mathbb{R}_4[x]\\). Por tanto, la norma que buscamos ser\u00e1:<\/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\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">float<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">integrate<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/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_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[52.79\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Sea el subespacio vectorial de \\(S\\subset\\mathbb{R}^5\\) generado por los vectores \\(\\vec{u}\\)(-11,-3,3,5,-1), \\(\\vec{v}\\)(7,2,-2,-3,1) y \\(\\vec{w}\\)(-9,-2,2,5,1) y \\(\\vec{x}\\)(0,-1,1,-2,0). \u00bfCu\u00e1l es la \\(\\textbf{dim}(S^\\perp)\\)?<\/p><\/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<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGorto06.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(S=\\{[[3a+2b,-2a-b],[b,a]]\\in \\mathcal{M}_2(\\mathbb{R})\\}\\). \u00bfCu\u00e1l es la \\(\\|\\textbf{proy}_S([[-1,0],[2,1]])\\|\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3() {\n  var htmlShow3 = document.getElementById(\"html-show3\");\n  if (htmlShow3.style.display === \"none\") {\n    htmlShow3.style.display = \"block\";\n  } else {\n    htmlShow3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGproy03.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(\\pi:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+3y-z=0,\\ y+2z-t=0\\}\\). Determinar la norma de la proyecci\u00f3n de [0,2,1,-1] sobre \\(S^\\bot\\). <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGproy04b.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea  \\(S=\\{[[3a+2b,-2a-b+c],[b+2c,a-c]]\\in \\mathcal{M}_2(\\mathbb{R})\\}\\), \\(A\\) y \\(B\\) las proyecciones sobre \\(S\\) y su ortogonal, respectivamente, de la matriz [[-1,0],[2,1]], \u00bfcu\u00e1l es el valor \\(\\|AB\\|\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4b() {\n  var htmlShow4b = document.getElementById(\"html-show4b\");\n  if (htmlShow4b.style.display === \"none\") {\n    htmlShow4b.style.display = \"block\";\n  } else {\n    htmlShow4b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4b\" style=\"display: none;\">\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGproy05.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea  \\[A=\\begin{bmatrix}-1 &#038; 4 &#038; 3\\\\ 1 &#038; -2 &#038; 0\\\\ 1 &#038; -2 &#038; 2\\\\ 1 &#038; 0 &#038; 1\\end{bmatrix}.\\] Si \\(v:[-1,2,3]\\), \\(u:[-1,2,3,4]\\) y \\(L\\) es la pseudoinversa de A, \u00bfcu\u00e1nto es \\(v.L.u^t\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4b3() {\n  var htmlShow4b3 = document.getElementById(\"html-show4b3\");\n  if (htmlShow4b3.style.display === \"none\") {\n    htmlShow4b3.style.display = \"block\";\n  } else {\n    htmlShow4b3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4b3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4b3\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_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\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/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_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/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_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/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\">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>\\[\\operatorname{ }\\begin{bmatrix}-1 &amp; 4 &amp; 3\\\\1 &amp; -2 &amp; 0\\\\1 &amp; -2 &amp; 2\\\\1 &amp; 0 &amp; 1\\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\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">if <\/span><span class=\"code_function\">(rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">3)<\/span><span class=\"code_function\"> then <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">\u00abtiene pseudoinversa por la izquierda\u00bb<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">L<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">invert<\/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_operator\">)<\/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_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">L<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\">else<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">\u00abno tiene pseudoinversa por la izquierda\u00bb<\/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{tiene pseudoinversa por la izquierda}\\]\\[\\begin{bmatrix}-\\frac{1}{12} &amp; \\frac{1}{4} &amp; -\\frac{5}{12} &amp; \\frac{13}{12}\\\\\\frac{1}{12} &amp; 0 &amp; -\\frac{1}{3} &amp; \\frac{5}{12}\\\\\\frac{1}{6} &amp; 0 &amp; \\frac{1}{3} &amp; -\\frac{1}{6}\\end{bmatrix}\u00bb \u00ab\\]<\/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_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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">L<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-2\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea  \\[A=\\begin{bmatrix}1 &#038; 1\\\\<br \/>\n1 &#038; 0\\end{bmatrix}=QR\\] la factorizaci\u00f3n QR de la matriz \\(A\\) y \\(u:[2,1]\\), \u00bfcu\u00e1nto es \\(u.Q.u^t\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4b31() {\n  var htmlShow4b31 = document.getElementById(\"html-show4b31\");\n  if (htmlShow4b31.style.display === \"none\") {\n    htmlShow4b31.style.display = \"block\";\n  } else {\n    htmlShow4b31.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4b31()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4b31\" style=\"display: none;\">\nPara resolver el ejercicio vamos a utilizar el M\u00e9todo de Gram-Schmidt:<\/p>\n<p>Sean \\(\\mathbf{a}_1\\) y \\(\\mathbf{a}_2\\) las columnas de \\(A\\):<br \/>\n\\[ \\mathbf{a}_1 = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}, \\quad \\mathbf{a}_2 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\]<\/p>\n<p>Paso 1: Aplicar el proceso de Gram-Schmidt para obtener las columnas ortogonales \\(\\mathbf{v}_1\\) y \\(\\mathbf{v}_2\\).<\/p>\n<p>1.  Vector \\(\\mathbf{v}_1\\):<br \/>\n    \\[ \\mathbf{v}_1 = \\mathbf{a}_1 = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} \\]<\/p>\n<p>2.  Vector \\(\\mathbf{v}_2\\):<br \/>\n    \\[ \\mathbf{v}_2 = \\mathbf{a}_2 &#8211; \\text{proy}_{\\mathbf{v}_1} \\mathbf{a}_2 = \\mathbf{a}_2 &#8211; \\frac{\\mathbf{a}_2 \\cdot \\mathbf{v}_1}{\\|\\mathbf{v}_1\\|^2} \\mathbf{v}_1 \\]<br \/>\n    Calculamos los t\u00e9rminos:<\/p>\n<ul>\n<li>\\(\\mathbf{a}_2 \\cdot \\mathbf{v}_1 = (1)(1) + (0)(1) = 1\\)<\/li>\n<li>\\(\\|\\mathbf{v}_1\\|^2 = 1^2 + 1^2 = 2\\)<\/li>\n<\/ul>\n<p>    Sustituimos:<br \/>\n    \\[ \\mathbf{v}_2 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} &#8211; \\frac{1}{2} \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 1 &#8211; 1\/2 \\\\ 0 &#8211; 1\/2 \\end{bmatrix} = \\begin{bmatrix} 1\/2 \\\\ -1\/2 \\end{bmatrix} \\]<\/p>\n<p>Paso 2: Normalizar los vectores \\(\\mathbf{v}_1\\) y \\(\\mathbf{v}_2\\) para formar la matriz \\(Q\\).<\/p>\n<p>1.  Vector \\(\\mathbf{q}_1\\):<br \/>\n    \\[ \\|\\mathbf{v}_1\\| = \\sqrt{2} \\]<br \/>\n    \\[ \\mathbf{q}_1 = \\frac{\\mathbf{v}_1}{\\|\\mathbf{v}_1\\|} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 1\/\\sqrt{2} \\\\ 1\/\\sqrt{2} \\end{bmatrix} \\]<\/p>\n<p>2.  Vector \\(\\mathbf{q}_2\\):<br \/>\n    \\[ \\|\\mathbf{v}_2\\| = \\sqrt{(1\/2)^2 + (-1\/2)^2} = \\sqrt{1\/4 + 1\/4} = \\sqrt{1\/2} = \\frac{1}{\\sqrt{2}} \\]<br \/>\n    \\[ \\mathbf{q}_2 = \\frac{\\mathbf{v}_2}{\\|\\mathbf{v}_2\\|} = \\sqrt{2} \\begin{bmatrix} 1\/2 \\\\ -1\/2 \\end{bmatrix} = \\begin{bmatrix} \\sqrt{2}\/2 \\\\ -\\sqrt{2}\/2 \\end{bmatrix} = \\begin{bmatrix} 1\/\\sqrt{2} \\\\ -1\/\\sqrt{2} \\end{bmatrix} \\]<\/p>\n<p>La matriz ortogonal \\(Q\\) es:<br \/>\n\\[ Q = \\begin{bmatrix} \\mathbf{q}_1 &#038; \\mathbf{q}_2 \\end{bmatrix} = \\begin{bmatrix} 1\/\\sqrt{2} &#038; 1\/\\sqrt{2} \\\\ 1\/\\sqrt{2} &#038; -1\/\\sqrt{2} \\end{bmatrix} \\]<\/p>\n<p>Paso 3: Calcular la matriz triangular superior \\(R\\).<\/p>\n<p>La matriz \\(R\\) se calcula como \\(R = Q^T A\\):<br \/>\n\\[ R = Q^T A = \\begin{bmatrix} 1\/\\sqrt{2} &#038; 1\/\\sqrt{2} \\\\ 1\/\\sqrt{2} &#038; -1\/\\sqrt{2} \\end{bmatrix} \\begin{bmatrix} 1 &#038; 1 \\\\ 1 &#038; 0 \\end{bmatrix} \\]<\/p>\n<p>Los elementos de \\(R\\) son:<\/p>\n<ul>\n<li>\\(r_{11} = \\mathbf{q}_1 \\cdot \\mathbf{a}_1 = 1\/\\sqrt{2} + 1\/\\sqrt{2} = 2\/\\sqrt{2} = \\sqrt{2}\\)<\/li>\n<li>\\(r_{12} = \\mathbf{q}_1 \\cdot \\mathbf{a}_2 = 1\/\\sqrt{2} + 0 = 1\/\\sqrt{2}\\)<\/li>\n<li>\\(r_{21} = \\mathbf{q}_2 \\cdot \\mathbf{a}_1 = 1\/\\sqrt{2} &#8211; 1\/\\sqrt{2} = 0\\)<\/li>\n<li>\\(r_{22} = \\mathbf{q}_2 \\cdot \\mathbf{a}_2 = 1\/\\sqrt{2} &#8211; 0 = 1\/\\sqrt{2}\\)<\/li>\n<\/ul>\n<p>La matriz triangular superior \\(R\\) es:<br \/>\n\\[ R = \\begin{bmatrix} \\sqrt{2} &#038; 1\/\\sqrt{2} \\\\ 0 &#038; 1\/\\sqrt{2} \\end{bmatrix} \\]<\/p>\n<p>Para terminar<br \/>\n\\[[2,1]\\begin{bmatrix} 1\/\\sqrt{2} &#038; 1\/\\sqrt{2} \\\\ 1\/\\sqrt{2} &#038; -1\/\\sqrt{2} \\end{bmatrix}.[2,1]^t=\\frac{7}{\\sqrt{2}}\\approx 4.9497\\]\n<\/p><\/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%\">\n<p><strong>Ejercicio:<\/strong> Sea la matriz \\(A=\\begin{bmatrix}-4 &#038; 0\\\\-4 &#038; -7\\end{bmatrix}\\), \u00bfcu\u00e1l es el valor de \\(\\log(\\|A^9\\|)\\)<\/p>\n<div id=\"menu-a\">\n<ul>\n<li>18.01<\/li>\n<li>22.54<\/li>\n<li>31.48<\/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><br \/>\n<button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>C.)<\/strong><\/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\">(%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\">4<\/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_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/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\">2<\/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}-4 &amp; 0\\\\-4 &amp; -7\\end{bmatrix}\\]<\/p>\n<p>\\[\\left( x+4\\right) \\, \\left( x+7\\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\">(%i11) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/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_operator\">)<\/span><span class=\"code_endofline\">$<\/span> \u00a0 <span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/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\">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_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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">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\">4<\/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\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">D<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">7<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/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_number\">2<\/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>\\[\\]\\[solve: dependent equations eliminated: (2)\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\begin{bmatrix}-7 &amp; 0\\\\0 &amp; -4\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}0 &amp; -\\frac{3}{4}\\\\1 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Calcular la potencia 9 ser\u00e1<\/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\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A9<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">D<\/span><span class=\"code_operator\">^<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">9<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}-262144 &amp; 0\\\\-53455284 &amp; -40353607\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora la norma y su logaritmo<\/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\">(%i15) <\/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\">4<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">mat_trace<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A9<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A9<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">log<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[6.697\\cdot {{10}^{7}}\\]<\/p>\n<p>\\[18.01\\]<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Ejercicio: Sea la matriz \\(A=\\begin{bmatrix}4 &#038; 2 &#038; 0 &#038; 0\\\\ 3 &#038; 3 &#038; 0 &#038; 0\\\\ 0 &#038; 0 &#038; 2 &#038; 5\\\\ 0 &#038; 0 &#038; 0 &#038; 2\\end{bmatrix}\\)&#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-730","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/730","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=730"}],"version-history":[{"count":7,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/730\/revisions"}],"predecessor-version":[{"id":733,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/730\/revisions\/733"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=730"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=730"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}