{"id":679,"date":"2025-12-05T10:25:49","date_gmt":"2025-12-05T09:25:49","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=679"},"modified":"2025-12-05T16:39:03","modified_gmt":"2025-12-05T15:39:03","slug":"alg-el-espacio-vectorial-euclideo-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=679","title":{"rendered":"ALG: El espacio vectorial eucl\u00eddeo  con maxima"},"content":{"rendered":"<h2>Suma e intersecci\u00f3n de subespacios<\/h2>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean los subespacios vectoriales \\(S=\\textbf{Gen}\\{[[1,2],[2,1]],\\) \\([[0,-1],[1,1]]\\}\\) \\(\\in\\mathcal{M}_2(\\mathbb{R})\\) y \\(T=\\textbf{Gen}\\{[[-1,0],[3,-1]],\\)\\([[1,9],[9,-2]]\\}\\in\\mathcal{M}_2(\\mathbb{R})\\). Determinar las ecuaciones impl\u00edcitas de \\(S\\cap T\\).<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1g() {\n  var htmlShow1g = document.getElementById(\"html-show1g\");\n  if (htmlShow1g.style.display === \"none\") {\n    htmlShow1g.style.display = \"block\";\n  } else {\n    htmlShow1g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1g\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGinterseccion01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Dados los subespacios anteriores, \u00bfcu\u00e1les son las sumas(en valor absoluto) de las coordenadas de las componentes normales de las ecuaciones impl\u00edcitas que definen \\(S+T\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g() {\n  var htmlShow2g = document.getElementById(\"html-show2g\");\n  if (htmlShow2g.style.display === \"none\") {\n    htmlShow2g.style.display = \"block\";\n  } else {\n    htmlShow2g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsuma01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>Veamos un problema de incidencia y paralelismo:<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean \\(S=\\left\\{\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix}+\\left.\\begin{bmatrix}a+b&#038;3a-b\\\\ b&#038; -a\\end{bmatrix}\\right|a,b\\in\\mathbb{R}\\right\\}\\) y \\(T=\\left\\{\\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3\\end{bmatrix}+\\left.\\begin{bmatrix}d&#038;c-2d\\\\ -c&#038; d+c\\end{bmatrix}\\right|c,d\\in\\mathbb{R}\\right\\}\\) \u00bfCu\u00e1l es su posici\u00f3n relativa?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1b() {\n  var htmlShow1b = document.getElementById(\"html-show1b\");\n  if (htmlShow1b.style.display === \"none\") {\n    htmlShow1b.style.display = \"block\";\n  } else {\n    htmlShow1b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1b\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGinters02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>El espacio vectorial eucl\u00eddeo<\/h2>\n<blockquote><p><strong>Ejemplo:<\/strong> Calcular \\(\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix}\\bullet\\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3\\end{bmatrix}\\) <\/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\/EjrALGpescalar01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Calcular \\(\\left( -3 {{x}^{2}}+2 x+1\\right) \\bullet \\left( {{x}^{2}}-x-2\\right)\\) <\/p><\/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\/EjrALGpescalar02.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es el coseno entre los vectores \\(x-x^2\\) y \\(1+x+x^2\\)? <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3g() {\n  var htmlShow3g = document.getElementById(\"html-show3g\");\n  if (htmlShow3g.style.display === \"none\") {\n    htmlShow3g.style.display = \"block\";\n  } else {\n    htmlShow3g.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3g\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/2021\/11\/Ejer_producto_escalar.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Calcular \\(\\cos\\left(\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix},\\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3\\end{bmatrix}\\right)\\) <\/p><\/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\/EjrALGcoseno01.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<blockquote><p><strong>Ejemplo:<\/strong> Calcular \\(\\text{dist}\\left(\\begin{bmatrix}1&#038;2\\\\ 0&#038; -1\\end{bmatrix},\\begin{bmatrix}0&#038;-1\\\\ 1&#038; 3\\end{bmatrix}\\right)\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGdist02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>Factorizci\u00f3n QR<\/h2>\n<blockquote><p><strong>Ejercicio:<\/strong> Encontrar la factorizaci\u00f3n QR de la matriz \\[\\begin{bmatrix}1 &#038; 2\\\\<br \/>\n-1 &#038; 3\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv253() {\n  var htmlShow253 = document.getElementById(\"html-show253\");\n  if (htmlShow253.style.display === \"none\") {\n    htmlShow253.style.display = \"block\";\n  } else {\n    htmlShow253.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv253()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show253\" 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\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/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\">0<\/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_number\">0<\/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}1 &amp; 2\\\\-1 &amp; 3\\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\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">a1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">col<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">a2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">col<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ 1{,}-1\\right] \\]<\/p>\n<p>\\[{ }\\left[ 2{,}3\\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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/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_operator\">:<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a1<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">a1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\sqrt{2}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">q1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">R<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\frac{1}{\\sqrt{2}}{,}-\\frac{1}{\\sqrt{2}}\\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\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-\\frac{1}{\\sqrt{2}}\\]<\/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\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a2<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\frac{5}{2}{,}\\frac{5}{2}\\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\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/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_operator\">:<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\frac{5}{\\sqrt{2}}\\]<\/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\">q2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\frac{1}{\\sqrt{2}}{,}\\frac{1}{\\sqrt{2}}\\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\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">Q<\/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\">q1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">q2<\/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}\\frac{1}{\\sqrt{2}} &amp; \\frac{1}{\\sqrt{2}}\\\\-\\frac{1}{\\sqrt{2}} &amp; \\frac{1}{\\sqrt{2}}\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Verifiquemos el resultado<\/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\">(%i13) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">Q<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">R<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 2\\\\-1 &amp; 3\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determina la factorizaci\u00f3n QR de la matriz \\(\\begin{bmatrix}1 &#038; 1 &#038; 0 \\\\<br \/>\n1 &#038; 0 &#038; 1 \\\\<br \/>\n0 &#038; 1 &#038; 1\\end{bmatrix}\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv9a1h() {\n  var htmlShow9a1h = document.getElementById(\"html-show9a1h\");\n  if (htmlShow9a1h.style.display === \"none\") {\n    htmlShow9a1h.style.display = \"block\";\n  } else {\n    htmlShow9a1h.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv9a1h()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show9a1h\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">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\">0<\/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_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">col<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/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_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\left[ 1{,}1{,}0\\right] {,}\\left[ 1{,}0{,}1\\right] {,}\\left[ 0{,}1{,}1\\right] \\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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v1<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">R<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\frac{1}{\\sqrt{2}}{,}\\frac{1}{\\sqrt{2}}{,}0\\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\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v2<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ \\frac{1}{\\sqrt{2} \\sqrt{3}}{,}-\\frac{1}{\\sqrt{2} \\sqrt{3}}{,}\\frac{\\sqrt{2}}{\\sqrt{3}}\\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\">(%i15) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">R<\/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_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">q2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">q2<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v3<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ -\\frac{1}{\\sqrt{3}}{,}\\frac{1}{\\sqrt{3}}{,}\\frac{1}{\\sqrt{3}}\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i17) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">q2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">q3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">\u00ab=\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">\u00ab.\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">R<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">\u00ab=\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{align}\\begin{bmatrix}1 &amp; 1 &amp; 0\\\\1 &amp; 0 &amp; 1\\\\0 &amp; 1 &amp; 1\\end{bmatrix}&#038;=\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &amp; \\frac{1}{\\sqrt{2} \\sqrt{3}} &amp; -\\frac{1}{\\sqrt{3}}\\\\\\frac{1}{\\sqrt{2}} &amp; -\\frac{1}{\\sqrt{2} \\sqrt{3}} &amp; \\frac{1}{\\sqrt{3}}\\\\0 &amp; \\frac{\\sqrt{2}}{\\sqrt{3}} &amp; \\frac{1}{\\sqrt{3}}\\end{bmatrix}\\cdot\\begin{bmatrix}\\sqrt{2} &amp; \\frac{1}{\\sqrt{2}} &amp; \\frac{1}{\\sqrt{2}}\\\\0 &amp; \\frac{\\sqrt{3}}{\\sqrt{2}} &amp; \\frac{1}{\\sqrt{2} \\sqrt{3}}\\\\0 &amp; 0 &amp; \\frac{2}{\\sqrt{3}}\\end{bmatrix}\\\\ &#038;=\\begin{bmatrix}1 &amp; 1 &amp; 0\\\\1 &amp; 0 &amp; 1\\\\0 &amp; 1 &amp; 1\\end{bmatrix}\\end{align}\\]<\/p>\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> \u00bfCu\u00e1l es \\(\\text{dist}\\left( -3 {{x}^{2}}+2 x+1,{{x}^{2}}-x-2\\right)\\)?<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>3.19<\/li>\n<li>5.65<\/li>\n<li>7.24<\/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><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGdist03.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Suma e intersecci\u00f3n de subespacios Ejemplo: Sean los subespacios vectoriales \\(S=\\textbf{Gen}\\{[[1,2],[2,1]],\\) \\([[0,-1],[1,1]]\\}\\) \\(\\in\\mathcal{M}_2(\\mathbb{R})\\) y \\(T=\\textbf{Gen}\\{[[-1,0],[3,-1]],\\)\\([[1,9],[9,-2]]\\}\\in\\mathcal{M}_2(\\mathbb{R})\\). Determinar las ecuaciones impl\u00edcitas de \\(S\\cap T\\). Soluci\u00f3n: Ejemplo: Dados los subespacios anteriores, \u00bfcu\u00e1les son las sumas(en&#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":[7],"class_list":["post-679","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/679","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=679"}],"version-history":[{"count":4,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/679\/revisions"}],"predecessor-version":[{"id":701,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/679\/revisions\/701"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=679"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=679"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=679"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}