{"id":452,"date":"2026-10-28T08:15:25","date_gmt":"2026-10-28T07:15:25","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=452"},"modified":"2026-10-25T18:06:48","modified_gmt":"2026-10-25T17:06:48","slug":"alg-ejercicios-de-repaso","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=452","title":{"rendered":"ALG: Ejercicios de repaso"},"content":{"rendered":"<h2>N\u00facleo e imagen de una aplicaci\u00f3n lineal<\/h2>\n<p>Veamos c\u00f3mo utilizamos maxima para calcular el n\u00facleo e imagen de una aplicaci\u00f3n lineal.<\/p>\n<p>Recordemos es dada una aplicaci\u00f3n lineal, \\(T\\), se define el <b>n\u00facleo<\/b> (ker) y la <b>imagen<\/b> (Im) de \\(T:V\\to W\\) como:<\/p>\n<blockquote>\n<dl>\n<dd>\\(\\mathbf{ker}(T)=\\{\\,v\\in V:T(v)=0_W\\,\\}\\)<\/dd>\n<dd>\\(\\mathbf{Im}(T)=\\{\\,w\\in W: \\exists v\\in V:T(v)=w\\,\\}\\)<\/dd>\n<\/dl>\n<\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Consideremos la aplicaci\u00f3n lineal \\(f : \\mathbb{R}^3\\to\\mathbb{R}^2\\) definida por \\[f(x, y,z) = [x-2*y+z,x+y-3*z].\\] Sea \\(\\mathbf{u}\\) el vector unitario director del \\(\\mathbf{ker}(f)\\). \u00bfCu\u00e1l es el valor de \\([2,-1,3].\\mathbf{u}\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w() {\n  var htmlShow1w = document.getElementById(\"html-show1w\");\n  if (htmlShow1w.style.display === \"none\") {\n    htmlShow1w.style.display = \"block\";\n  } else {\n    htmlShow1w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w\" 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\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ x=\\frac{5 {\\mathrm{\\% r1}}}{3}\\operatorname{,}y=\\frac{4 {\\mathrm{\\% r1}}}{3}\\operatorname{,}z={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/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\">3<\/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_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }2.121320343559642\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Consideremos la aplicaci\u00f3n lineal \\(f : \\mathbb{R}^3\\to\\mathbb{R}^2\\) definida por \\[f(x, y,z) = [x-2*y+z,x+y-3*z].\\] Obtener una base de \\(\\mathbf{Im}(f)\\).<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w2() {\n  var htmlShow1w2 = document.getElementById(\"html-show1w2\");\n  if (htmlShow1w2.style.display === \"none\") {\n    htmlShow1w2.style.display = \"block\";\n  } else {\n    htmlShow1w2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1w2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w2\" 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\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/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_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/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_variable\">v<\/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\">v<\/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_variable\">v<\/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_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">I<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><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_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 1\\\\-2 &amp; 1\\\\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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }2\\]<\/p>\n<p>Como el rango es 2 solo necesitamos dos filas de la matriz \\(A\\) linealmente independientes para constituir una base de \\(\\mathbf{Im}(f)\\)\n<\/p><\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea la aplicaci\u00f3n lineal \\(f:\\mathbb{R}^3\\to\\mathcal{M}_2(\\mathbb{R})\\) dada por \\[f(x,y,z)=<br \/>\n\\begin{bmatrix}x-y&#038;y-z\\\\ z-x&#038;x-y\\end{bmatrix}.\\] \u00bfCu\u00e1nto es \\(\\mathbf{dim}\\,\\mathbf{Ker}(f)+3\\,\\mathbf{dim}\\,\\mathbf{Im}(f)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w3() {\n  var htmlShow1w3 = document.getElementById(\"html-show1w3\");\n  if (htmlShow1w3.style.display === \"none\") {\n    htmlShow1w3.style.display = \"block\";\n  } else {\n    htmlShow1w3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1w3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w3\" style=\"display: none;\">\nRecordemos que podemos establecer un isomorfismo \\(\\Phi :\\mathcal{M}_2(\\mathbb{R})\\to\\mathbb{R}^4\\) dado por \\[\\Phi\\begin{bmatrix}a&#038;b\\\\ c&#038;d\\end{bmatrix}=[a,b,c,d].\\]<br \/>\nPor tanto, podemos usar este isomorfismo para trabajar con vectores en vez de matrices.<br \/>\n <!-- 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_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/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_operator\">\u2212<\/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_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\">\u2212<\/span><span class=\"code_variable\">v<\/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\">v<\/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\">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\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/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_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">I<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><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_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; -1 &amp; 1\\\\-1 &amp; 1 &amp; 0 &amp; -1\\\\0 &amp; -1 &amp; 1 &amp; 0\\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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }2\\]<\/p>\n<p>Como el rango es 2, nos dice que \\(\\mathbf{Im}(f)=2\\). Si recordamos que \\[dim\\,\\mathbf{Ker}(f) + dim\\,\\mathbf{Im}(f)=dim\\,\\mathbb{R}^3 \\] tendremos que \\(\\mathbf{Ker}(f)=1\\), luego \\[\\mathbf{dim}\\,\\mathbf{Ker}(f)+3\\,\\mathbf{dim}\\,\\mathbf{Im}(f)=1+3\\cdot 2=7\\]<\/p>\n<p>Para terminar, observemos que una base del conjunto imagen la obtenemos con dos filas linealmente independientes de \\(A\\). Si consideramos las dos \u00faltimas, la matriz es escalonada, luego son l.i. As\u00ed la base la obtendr\u00edamos usando el isomorfismo \\(\\Phi ^{-1}\\); es decir, transformando las filas en matrices:<br \/>\n\\[\\left\\{<br \/>\n\\begin{bmatrix}-1 &#038; 1 \\\\ 0 &#038; -1\\end{bmatrix},<br \/>\n\\begin{bmatrix}0 &#038; -1 \\\\ 1 &#038; 0\\end{bmatrix}<br \/>\n\\right\\}\\]\n<\/p><\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Sean las aplicaciones lineales \\(f:\\mathbb{R}^3\\to\\mathbb{R}_{3}[X]\\) dada por \\[f(a,b,c)=a+(b-a)X+(c-a)X^2+(2a-b)X^3,\\] y \\(g:\\mathbb{R}_3[X]\\to\\mathcal{M}_2(\\mathbb{R})\\) dada por \\[g(p_0+p_1X+p_2X^2+p_3X^3)= \\begin{bmatrix}p_0&#038;p_2-p_1\\\\ p_1-p_2&#038;p_3\\end{bmatrix}.\\] \u00bfCu\u00e1l es el valor del determinante de \\((g\\circ f)(-1,3,1)\\)?\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1cs() {\n  var htmlShow1cs = document.getElementById(\"html-show1cs\");\n  if (htmlShow1cs.style.display === \"none\") {\n    htmlShow1cs.style.display = \"block\";\n  } else {\n    htmlShow1cs.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1cs()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1cs\" style=\"display: none;\">\nRecordad que para trabajar con mayor comodidad trabajaremos con los isomorfismos \\(\\Phi:\\mathbb{R}_{3}[X]\\to\\mathbb{R}^4\\), dado por \\(\\Phi(p_0+p_1X+p_2X^2+p_3X^3)=[p_0,p_1,p_2,p_3]\\), y el \\(\\Psi:\\mathcal{M}_2(\\mathbb{R})\\to\\mathbb{R}^4\\), dado por \\(\\Psi\\begin{bmatrix}a&#038;b\\\\ c&#038;d\\end{bmatrix}=[a,b,c,d]\\)<\/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\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/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\">\u2212<\/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\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/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_number\">2<\/span><span class=\"code_operator\">\u00b7<\/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_operator\">\u2212<\/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_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">g<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/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\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/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_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\">\u2212<\/span><span class=\"code_variable\">v<\/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\">v<\/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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{f}(v)\\operatorname{:=}\\left[ {v_1}\\operatorname{,}{v_2}-{v_1}\\operatorname{,}{v_3}-{v_1}\\operatorname{,}2 {v_1}-{v_2}\\right] \\]<\/p>\n<p>\\[\\operatorname{g}(v)\\operatorname{:=}\\left[ {v_1}\\operatorname{,}{v_3}-{v_2}\\operatorname{,}{v_2}-{v_3}\\operatorname{,}{v_4}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora calculamos las matrices asociadas a cada aplicaci\u00f3n:<\/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=\"code_variable\">I<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">lt<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">Mf<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">lt<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\-1 &amp; 1 &amp; 0\\\\-1 &amp; 0 &amp; 1\\\\2 &amp; -1 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">I<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><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_function\">g<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/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_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">lt<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">g<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">row<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">I<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">Mg<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">lt<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/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; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; 1 &amp; 0\\\\0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La matriz de la composic\u00f3n ser\u00e1 el producto de las matrices:<\/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\">(%i11) <\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">Mg<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">Mf<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; 1\\\\0 &amp; 1 &amp; -1\\\\2 &amp; -1 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora solo tenemos que multiplicarlo por el vector:<\/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\">Mg<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">Mf<\/span><span class=\"code_endofline\">.<\/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\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/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>\\[\\begin{bmatrix}-1\\\\-2\\\\2\\\\-5\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Estas son las coordenadas de la matriz resultados. Si lo pasamos a matriz, tendremos<\/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\">s<\/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_operator\">\u2212<\/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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/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\\\\2 &amp; -5\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para resolver el ejercicio, calculamos su determinante:<\/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\">(%i14) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[9\\]<\/p>\n<\/div>\n<hr \/>\n<h3>El plano af\u00edn \\(\\mathbb{R}^2\\)<\/h2>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar si los puntos P(4,1), Q(3,-4) y R(2,-1) son colineales.<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1io() {\n  var htmlShow1io = document.getElementById(\"html-show1io\");\n  if (htmlShow1io.style.display === \"none\") {\n    htmlShow1io.style.display = \"block\";\n  } else {\n    htmlShow1io.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1io()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1io\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin03.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  \u00bfCu\u00e1nto suman los coeficientes que multiplican a \\(x\\) e \\(y\\)  en la ecuaci\u00f3n impl\u00edcita de la recta del plano af\u00edn que pasa por los puntos P(4,1) y Q(7,-4)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2io() {\n  var htmlShow2io = document.getElementById(\"html-show2io\");\n  if (htmlShow2io.style.display === \"none\") {\n    htmlShow2io.style.display = \"block\";\n  } else {\n    htmlShow2io.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2io()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2io\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<h3>El espacio af\u00edn \\(\\mathbb{R}^3\\)<\/h2>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar las ecuaciones param\u00e9tricas de la recta que pasa por los puntos P(4,1,2) y Q(3,-4,0)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3f() {\n  var htmlShow3f = document.getElementById(\"html-show3f\");\n  if (htmlShow3f.style.display === \"none\") {\n    htmlShow3f.style.display = \"block\";\n  } else {\n    htmlShow3f.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3f()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3f\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin04.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar las ecuaciones impl\u00edcitas de la recta que pasa por los puntos P(4,1,2) y Q(3,-4,0)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3w() {\n  var htmlShow3w = document.getElementById(\"html-show3w\");\n  if (htmlShow3w.style.display === \"none\") {\n    htmlShow3w.style.display = \"block\";\n  } else {\n    htmlShow3w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3w\" 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=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">4<\/span>,<span class=\"code_number\">0<\/span>]<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Dados estos dos puntos la recta vendr\u00e1 dada por uno de ellos, por ejemplo P, y el vector director obtenido de la resta de ambos, \\(\\overrightarrow{PQ}\\)<\/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><span class=\"input\"><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>As\u00ed, las ecuaciones impl\u00edcitas las determinaran las equaciones que impliquen rango igual a 1 en la matriz:<\/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=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span>(<span class=\"code_function\">matrix<\/span>(<span class=\"code_variable\">r<\/span>,[<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/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=\"center\"><mtd><mtext>(A)<\/mtext><\/mtd><mtd><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><mtd><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>5<\/mn><\/mrow><\/mtd><mtd><mrow><mi>y<\/mi><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><mtd><mrow><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Para conseguir estas ecuaciones, consideremas un menor de orden 1 distinto de cero y consigamos escalonar la matriz en base a dicho menor:<\/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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/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=\"center\"><mtd><mtext\/><\/mtd><mtd><mo>0 errores, 0 advertencias<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(A)<\/mtext><\/mtd><mtd><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><mtd><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mrow><mi>y<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>\u2062<\/mo><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><mo>)<\/mo><\/mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mrow><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><mo>)<\/mo><\/mrow><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Las ecuaciones que buscamos se corresponde con las ecuaciones que hacen cero las dos lineas por debajo del menor escogido:<\/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\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_function\">rat<\/span>(<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>])<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">rat<\/span>(<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">2<\/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=\"center\"><mtd><mtext>(%o7)\/R\/ <\/mtext><\/mtd><mtd><mi>y<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>19<\/mn><mi>=<\/mi><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(%o8)\/R\/ <\/mtext><\/mtd><mtd><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mi>=<\/mi><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar la ecuaci\u00f3n impl\u00edcita del plano que pasa por el punto P(4,1,2) y tiene por vectores directores \\(\\vec{v}(1,5,3)\\) y \\(\\vec{u}(3,-2,1)\\)<\/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\/EjrALGrect_afin05.html\" width=\"650\" height=\"210\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea S(\\(x\\),4,-3), \u00bfexiste alg\u00fan valor de \\(x\\) que hace a S coplanario con P(0,-2,1), Q(1,-2,-3) y R(1,-3,1)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4a() {\n  var htmlShow4a = document.getElementById(\"html-show4a\");\n  if (htmlShow4a.style.display === \"none\") {\n    htmlShow4a.style.display = \"block\";\n  } else {\n    htmlShow4a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a\" 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\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">S<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pr<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">ps<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">S<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>(<span class=\"code_variable\">pq<\/span>,<span class=\"code_variable\">pr<\/span>,<span class=\"code_variable\">ps<\/span>)))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">x<\/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>(eq)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>4<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>20<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o9) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<h2>El espacio af\u00edn eucl\u00eddeo<\/h2>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean \\(\\vec{v}=(1,-2,3)\\) y  \\(\\vec{u}=(2,1,-1)\\), calcular \\(\\vec{v}\\times\\vec{u}\\).<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1yu() {\n  var htmlShow1yu = document.getElementById(\"html-show1yu\");\n  if (htmlShow1yu.style.display === \"none\") {\n    htmlShow1yu.style.display = \"block\";\n  } else {\n    htmlShow1yu.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1yu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1yu\" style=\"display: none;\">\n\\[\\begin{vmatrix} \\vec{i} &#038; \\vec{j} &#038;\\vec{k}\\\\ 1&#038;-2&#038;3 \\\\ 2&#038;1&#038;-1\\end{vmatrix}=\\begin{vmatrix}-2&#038;3\\\\ 1&#038;-1\\end{vmatrix}\\vec{i} &#8211; \\begin{vmatrix}1&#038;3\\\\ 2&#038;-1\\end{vmatrix}\\vec{j} +\\begin{vmatrix}1&#038;-2\\\\ 2&#038;1\\end{vmatrix}\\vec{k}\\] \\[=-\\vec{i} +7\\vec{j} +5\\vec{k}\\]\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es, en valor absoluto, el producto escalar del vector \\([1,2,4]\\) por el vector normal unitario de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv81() {\n  var htmlShow81 = document.getElementById(\"html-show81\");\n  if (htmlShow81.style.display === \"none\") {\n    htmlShow81.style.display = \"block\";\n  } else {\n    htmlShow81.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv81()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show81\" style=\"display: none;\">\nEl vector normal de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\) vendr\u00e1 dado por el producto vectorial de los vectores normales de cada plano:<\/p>\n<p>\\[\\begin{vmatrix}\\mathbf{i}&#038;\\mathbf{j}&#038;\\mathbf{k}\\\\ 1&#038;9&#038;-6 \\\\ -4&#038;-18&#038;15\\end{vmatrix}=27\\mathbf{i}+9\\mathbf{j}+18\\mathbf{k}\\]<br \/>\nLuego el vector normal es [27,9,18]. Ahora lo normalizamos(hacerlo unitario) \\[\\frac{1}{\\|[27,9,18]\\|}[27,9,18]=\\left[ \\frac{3}{\\sqrt{14}},\\frac{1}{\\sqrt{14}},\\frac{2}{\\sqrt{14}}\\right]\\]<br \/>\nPor \u00faltimo,<br \/>\n\\[[1,2,4].\\left[ \\frac{3}{\\sqrt{14}},\\frac{1}{\\sqrt{14}},\\frac{2}{\\sqrt{14}}\\right]=\\frac{13}{\\sqrt{14}}\\approx 3.47\\]\n<\/p><\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Sean \\(\\vec{v}=(1,-2,3)\\) y  \\(\\vec{u}=(2,1,-1)\\), \u00bfcu\u00e1l es la segunda cifra decimal del \\(\\cos(\\vec{v},\\vec{u})\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2yu() {\n  var htmlShow2yu = document.getElementById(\"html-show2yu\");\n  if (htmlShow2yu.style.display === \"none\") {\n    htmlShow2yu.style.display = \"block\";\n  } else {\n    htmlShow2yu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2yu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2yu\" style=\"display: none;\">\nDados\\(\\vec{v}=(1,-2,3)\\) y \\(\\vec{u}=(2,1,-1)\\), \\(\\|(1,-2,3)\\|=\\sqrt{1^2+(-2)^2+3^2}=\\sqrt{14}\\), \\(\\|(2,1,-1)\\|=\\sqrt{6}\\). Luego \\[\\mathbf{cos(\\vec{v},\\vec{u})}=-\\frac{3}{\\sqrt{6} \\sqrt{14}}\\approx -0.3273\\]\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Cu\u00e1l es la proyecci\u00f3n de \\(\\vec{u}=(1,-2,3)\\) sobre \\(\\vec{v}=(2,1,-1)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2au() {\n  var htmlShow2au = document.getElementById(\"html-show2au\");\n  if (htmlShow2au.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2au.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2au()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2au\" style=\"display: none;\">\n\\[\\textbf{proy}_\\vec{v}(\\vec{u})=\\frac{(1,-2,3)\\bullet (2,1,-1)}{\\|(2,1,-1)\\|^2}(2,1,-1)=-\\frac{3}{6}(2,1,-1)\\]\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar la componente del vector \\([3,2,1]\\) en la direcci\u00f3n del vector \\([2,-1,0]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31cu() {\n  var htmlShow31cu = document.getElementById(\"html-show31cu\");\n  if (htmlShow31cu.style.display === \"none\") {\n    htmlShow31cu.style.display = \"block\";\n  } else {\n    htmlShow31cu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv31cu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31cu\" style=\"display: none;\">\n<strong>Soluci\u00f3n<\/strong>:<br \/>\nPor la definici\u00f3n tendremos: \\[\\textbf{comp}_{[2,-1,0]}([3,2,1])=\\frac{[3,2,1]\\bullet[2,-1,0]}{\\|[2,-1,0]\\|}=\\frac{4}{\\sqrt{5}}\\]\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la distancia entre los puntos \\(P(1,-1,2)\\) y \\(Q(0,2,-2)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2bu() {\n  var htmlShow2bu = document.getElementById(\"html-show2bu\");\n  if (htmlShow2bu.style.display === \"none\") {\n    htmlShow2bu.style.display = \"block\";\n  } else {\n    htmlShow2bu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2bu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2bu\" style=\"display: none;\">\n\\[[d(P,Q)=||\\vec{QP}||=\\sqrt{(0-1)^2 +(2-(-1))^2 + ((-2)-2)^2}=\\sqrt{26}\\]\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Cu\u00e1l es la distancia del punto \\(Q(0,2,-2)\\) a la recta \\(r:\\{(1,-1,2)+\\mathbf{Gen}\\{(3,2,-1)\\}\\}\\in \\mathbb{R}^3\\) ?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2cu() {\n  var htmlShow2cu = document.getElementById(\"html-show2cu\");\n  if (htmlShow2cu.style.display === \"none\") {\n    htmlShow2cu.style.display = \"block\";\n  } else {\n    htmlShow2cu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2cu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2cu\" style=\"display: none;\">\n\\[d(Q,r)=\\frac{\\|\\vec{PQ}\\times\\vec{v}\\|}{\\|(3,2,-1)\\|}=\\frac{\\|(5,-13,-11)\\|}{\\|(3,2,-1)\\|}= \\frac{3 \\sqrt{35}}{\\sqrt{14}}\\approx4.7434\\]<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Cu\u00e1l es la distancia del punto \\(Q(0,2,-2)\\) al plano \\(\\pi:\\{(1,-1,2)+\\mathbf{Gen}\\{(3,2,-1),(0,1,1)\\}\\}\\in \\mathbb{R}^3\\) ?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2d() {\n  var htmlShow2d = document.getElementById(\"html-show2d\");\n  if (htmlShow2d.style.display === \"none\") {\n    htmlShow2d.style.display = \"block\";\n  } else {\n    htmlShow2d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d\" style=\"display: none;\">\nLa ecuaci\u00f3n impl\u00edcita del plano vendr\u00e1 dada por: \\[\\begin{vmatrix}x-1 &#038; y+1 &#038; z-2\\\\<br \/>\n3 &#038; 2 &#038; -1\\\\<br \/>\n0 &#038; 1 &#038; 1\\end{vmatrix}=3z-3y+3x-12=0\\]<br \/>\n\\[d(Q,\\pi)=\\frac{|(3,-3,3).(0,2,-2)-12|}{\\|(3,-3,3)\\|}=\\frac{8}{\\sqrt{3}}\\approx 4.6188\\]<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Determina si O(2,4,-13) es coplanario con \\(P\\)(1,-3,-1), \\(Q\\)(2,-2,1) y \\(R\\)(3,2,-4)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx2() {\n  var htmlShow2dx2 = document.getElementById(\"html-show2dx2\");\n  if (htmlShow2dx2.style.display === \"none\") {\n    htmlShow2dx2.style.display = \"block\";\n  } else {\n    htmlShow2dx2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Puntos Coplanarios. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/9g6oZ805A5Y?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 S(-2,3,x). \u00bfQu\u00e9 valor de x hace que S sea coplanario con P(2,2,-1), Q(1,2,-3) y R(3,-2,1) <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx4() {\n  var htmlShow2dx4 = document.getElementById(\"html-show2dx4\");\n  if (htmlShow2dx4.style.display === \"none\") {\n    htmlShow2dx4.style.display = \"block\";\n  } else {\n    htmlShow2dx4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx4\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Puntos coplanarios. Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/LlkxBlb1PY0?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> Sean \\(\\vec{a}\\) y \\(\\vec{b}\\) dos vectores tales que \\(||\\vec{a}||=3\\), \\(||\\vec{b}||=\\sqrt{6}\\) y \\(\\theta=45\u00ba\\), \u00bfcu\u00e1l es su producto escalar?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx5() {\n  var htmlShow2dx5 = document.getElementById(\"html-show2dx5\");\n  if (htmlShow2dx5.style.display === \"none\") {\n    htmlShow2dx5.style.display = \"block\";\n  } else {\n    htmlShow2dx5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx5\" style=\"display: none;\">\nSabemos que \\(\\cos(45\u00ba)=\\cos(\\tfrac{\\pi}{4})=\\frac{1}{\\sqrt{2}}\\). Adem\u00e1s sabemos que \\[\\textbf{cos}(\\vec{v},\\vec{u})=\\frac{\\vec{v}\\bullet\\vec{u}}{||\\vec{v}||\\cdot ||\\vec{u}||}.\\] Luego, como el \u00e1ngulo de nuestro vectores es de \\(\\frac{\\pi}{\\sqrt{4}}\\), ser\u00e1 \\[\\frac{1}{\\sqrt{2}}=\\textbf{cos}(\\vec{a},\\vec{b})=\\frac{\\vec{a}\\bullet\\vec{b}}{3\\cdot \\sqrt{6}}\\Rightarrow \\vec{a}\\bullet\\vec{b}=\\frac{3\\sqrt{6}}{\\sqrt{2}}=3\\sqrt{3}\\]\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 de los puntos dados es coplanario con los puntos P(2,2,-1), Q(1,2,-3) y R(3,-2,1)? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>(-4,1,6)<\/li>\n<li>(0,6,-5)<\/li>\n<li>(-1,1,5)<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv(id) {\n  var htmlShow = document.getElementById(id);\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<p><em>La soluci\u00f3n se mostrar\u00e1 m\u00e1s adelante.<\/em><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>N\u00facleo e imagen de una aplicaci\u00f3n lineal Veamos c\u00f3mo utilizamos maxima para calcular el n\u00facleo e imagen de una aplicaci\u00f3n lineal. Recordemos es dada una aplicaci\u00f3n lineal, \\(T\\), se define el n\u00facleo&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-452","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/452","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=452"}],"version-history":[{"count":6,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/452\/revisions"}],"predecessor-version":[{"id":458,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/452\/revisions\/458"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=452"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=452"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}