{"id":378,"date":"2025-10-31T10:15:36","date_gmt":"2025-10-31T09:15:36","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=378"},"modified":"2025-10-31T16:50:37","modified_gmt":"2025-10-31T15:50:37","slug":"alg-aplicaciones-lineales-y-el-plano-afin-mathbbr2","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=378","title":{"rendered":"ALG: Aplicaciones lineales y el plano af\u00edn \\(\\mathbb{R}^2\\)"},"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 lo 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<h2>El plano af\u00edn \\(\\mathbb{R}^2\\)<\/h2>\n<p>En la pasada clase definimos el plano, veamos c\u00f3mo expresamos las variedades que contienen dichos espacios mediante sus ecuaciones impl\u00edcitas y param\u00e9tricas.<\/p>\n<p>Vimos que las ecuaciones param\u00e9tricas de una recta en el plano af\u00edn que pasa por un punto \\(P(p_1,p_2)\\) y que tiene por subespacio director el generado por el vector \\(\\vec{v}=(v_1,v_2)\\), vendr\u00e1 dada de la forma: \\[r=\\{(x,y)\\in\\mathbb{R}^2;(x,y)=(p_1,p_2)+\\lambda(v_1,v_2),\\lambda\\in\\mathbb{R}\\}\\]<\/p>\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 showHtmlDiv1() {\n  var htmlShow1 = document.getElementById(\"html-show1\");\n  if (htmlShow1.style.display === \"none\") {\n    htmlShow1.style.display = \"block\";\n  } else {\n    htmlShow1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin03.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>Un resultado m\u00e1s pr\u00e1ctico nos dice que la ecuaci\u00f3n impl\u00edcita de la recta en el plano af\u00edn que pasa por los puntos \\(P(p_1,p_2)\\) y \\(Q(q_1,q_2)\\) vendr\u00e1 dada por el determinante:<br \/>\n\\[\\begin{vmatrix} x &#038; y &#038; 1\\\\ p_1 &#038; p_2 &#038; 1\\\\ q_1 &#038; q_2 &#038; 1 \\end{vmatrix}=0\\]<\/p>\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 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\/EjrALGrect_afin02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<h2>El espacio af\u00edn \\(\\mathbb{R}^3\\)<\/h2>\n<p>En el espacio af\u00edn una recta que pasa por un punto \\(P(p_1,p_2,p_3)\\) y que tiene por subespacio director el generado por el vector \\(\\vec{v}=(v_1,v_2,v_3)\\), vendr\u00e1 dada de la forma: \\[r=\\{(x,y,z)\\in\\mathbb{R}^3;(x,y,z)=(p_1,p_2,p_3)+\\lambda(v_1,v_2,v_3),\\lambda\\in\\mathbb{R}\\}\\]<\/p>\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<p>Tambi\u00e9n podemos utilizar el resultado que nos dice que la ecuaci\u00f3n impl\u00edcita del plano en el espacio af\u00edn que pasa por un punto \\(P(p_1,p_2,p_3)\\) y que tiene por subespacio director el generado por los vectores \\(\\vec{v}=(v_1,v_2,v_3)\\) y \\(\\vec{u}=(u_1,u_2,u_3)\\), vendr\u00e1 determinado por el determinante \\[\\begin{vmatrix} x-p_1 &#038; y-p_2 &#038; z-p_3\\\\ v_1 &#038; v_2 &#038; v_3 \\\\ u_1 &#038; u_2 &#038; u_3 \\end{vmatrix}=0\\]<\/p>\n<\/p>\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<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() {\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>B.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin06.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\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":[6],"tags":[],"class_list":["post-378","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/378","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=378"}],"version-history":[{"count":16,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/378\/revisions"}],"predecessor-version":[{"id":417,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/378\/revisions\/417"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=378"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=378"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=378"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}