{"id":461,"date":"2026-11-02T08:12:54","date_gmt":"2026-11-02T07:12:54","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=461"},"modified":"2026-10-29T18:09:30","modified_gmt":"2026-10-29T17:09:30","slug":"alg-variedades-y-sistemas-de-ecuaciones","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=461","title":{"rendered":"ALG: Variedades y Sistemas de Ecuaciones"},"content":{"rendered":"<p>Ampliamos las definiciones de variedades lineales que, en muchos casos, las equiparamos con los subespacios vectoriales, aunque no tienen por qu\u00e9 serlos, a \\(\\mathbb{R}^n\\)<\/p>\n<p>Las variedades lineales nos dan pie para definir las ecuaciones param\u00e9tricas e impl\u00edcitas que las identifican.<\/p>\n<p>Adem\u00e1s, hemos introducido el <a href=\"http:\/\/es.wikipedia.org\/wiki\/Espacio_af%C3%ADn\">espacio af\u00edn<\/a> y con \u00e9l la variedad af\u00edn, una forma de dar sentido a las estructuras que conocemos de unir puntos con vectores. Ahora ya podemos hablar de rectas de puntos en el plano, o planos de puntos en el espacio.<\/p>\n<p>Como en el caso de las variedades lineales, podemos encontrar la <a href=\"https:\/\/es.wikipedia.org\/wiki\/Variedad_lineal\" target=\"_blank\" rel=\"noopener noreferrer\">variedad af\u00edn<\/a> definida por las ecuaciones param\u00e9tricas o impl\u00edcitas.<\/p>\n<h3>De param\u00e9tricas a impl\u00edcitas<\/h3>\n<p>Nuestra principal problema ser\u00e1 determinar las las ecuaciones param\u00e9tricas o impl\u00edcitas de una variedad. Veamos c\u00f3mo pasamos de las ecuaciones param\u00e9tricas a impl\u00edcitas. Consideremos una variedad dada por \\(\\pi:P+\\textbf{Gen}\\{\\vec{v}_1,\\vec{v}_2,\\ldots,\\vec{v}_r\\}\\). Las ecuaciones impl\u00edcitas nos las proporcionar\u00e1n las ecuaciones que hacen que \\[rang\\begin{bmatrix}v_{11}&#038; v_{21}&#038;\\cdots &#038;v_{r1} &#038;x_1-p_1\\\\ v_{12}&#038; v_{22}&#038;\\cdots &#038;v_{r2} &#038;x_2-p_2\\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots &#038;\\vdots\\\\ v_{1n}&#038; v_{2n}&#038;\\cdots &#038;v_{rn}&#038;x_r-p_r\\end{bmatrix}=r\\]<\/p>\n<p>Si en vez de una variedad tenemos un subespacio vectorial, el procedimiento ser\u00eda el mismo, como si el la ecuaci\u00f3n anterior las coordenadas \\(p_i=0\\):<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas del subespacio vectorial \\(\\textbf{r}=\\mathbf{Gen}\\left\\{(1,2,3)\\right\\}\\subset\\mathbb{R}^3\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1ds() {\n  var htmlShow1ds = document.getElementById(\"html-show1ds\");\n  if (htmlShow1ds.style.display === \"none\") {\n    htmlShow1ds.style.display = \"block\";\n  } else {\n    htmlShow1ds.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1ds()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1ds\" style=\"display: none;\">\nObservar que \\[\\textbf{rank}\\begin{bmatrix}1&#038;x\\\\ 2&#038;y \\\\ 3&#038;z\\end{bmatrix}=2\\]<br \/>\nY esto se cumple si<br \/>\n\\[\\begin{vmatrix}1&#038;x\\\\ 2&#038;y \\end{vmatrix}=y-2x=0,\\ \\begin{vmatrix}1&#038;x\\\\ 3&#038;z\\end{vmatrix}=z-3z=0\\]<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\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">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_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_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_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; x\\\\2 &amp; y\\\\3 &amp; z\\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\">(%i3)<\/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_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/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>\\[y-2 x=0\\]<\/p>\n<p>\\[z-3 x=0\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas del subespacio vectorial \\[\\textbf{T}=\\left\\{\\begin{bmatrix}a&#038;-b\\\\ b&#038;c\\end{bmatrix}\\in \\mathcal{M}_2(\\mathbb{R})\\right\\}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1d() {\n  var htmlShow1d = document.getElementById(\"html-show1d\");\n  if (htmlShow1d.style.display === \"none\") {\n    htmlShow1d.style.display = \"block\";\n  } else {\n    htmlShow1d.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1d\" style=\"display: none;\">\nObservar que se trata de verificar que \\[\\textbf{rank}\\begin{bmatrix}1&#038;0&#038;0&#038;a_{11}\\\\ 0&#038;-1&#038;0&#038;a_{12} \\\\ 0&#038;1&#038;0&#038;a_{21}\\\\ 0&#038;0&#038;1&#038;a_{22}\\end{bmatrix}=3\\]<br \/>\nY esto se cumple si<br \/>\n\\[\\begin{vmatrix}1&#038;0&#038;0&#038;a_{11}\\\\ 0&#038;-1&#038;0&#038;a_{12} \\\\ 0&#038;1&#038;0&#038;a_{21}\\\\ 0&#038;0&#038;1&#038;a_{22}\\end{vmatrix}=a_{12}+a_{21}=0\\]<br \/>\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Ecuaciones Impl\u00edcitas. Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/qAwlapL-3aU?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><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\">(%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\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">c<\/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\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">c<\/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\">transpose<\/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_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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 &amp; 0 &amp; 0 &amp; x\\\\0 &amp; -1 &amp; 0 &amp; y\\\\0 &amp; 1 &amp; 0 &amp; z\\\\0 &amp; 0 &amp; 1 &amp; t\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/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\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x\\\\0 &amp; -1 &amp; 0 &amp; y\\\\0 &amp; 0 &amp; 0 &amp; z+y\\\\0 &amp; 0 &amp; 1 &amp; t\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x\\\\0 &amp; -1 &amp; 0 &amp; y\\\\0 &amp; 0 &amp; 1 &amp; t\\\\0 &amp; 0 &amp; 0 &amp; z+y\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas del subespacio vectorial \\[\\textbf{T}=\\left\\{\\begin{bmatrix}a+2b&#038;a-b\\\\ b-a&#038;b\\end{bmatrix}\\in \\mathcal{M}_2(\\mathbb{R})\\right\\}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3d() {\n  var htmlShow3d = document.getElementById(\"html-show3d\");\n  if (htmlShow3d.style.display === \"none\") {\n    htmlShow3d.style.display = \"block\";\n  } else {\n    htmlShow3d.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3d\" style=\"display: none;\">\nObservar que se trata de verificar que \\[\\textbf{rank}\\begin{bmatrix}1 &#038; 2 &#038; x\\\\<br \/>\n1 &#038; -1 &#038; y\\\\<br \/>\n-1 &#038; 1 &#038; z\\\\<br \/>\n0 &#038; 1 &#038; t\\end{bmatrix}=2\\]<br \/>\nPara ello, es suficiente con que<br \/>\n\\[\\begin{vmatrix}1 &#038; 2 &#038; x\\\\<br \/>\n-1 &#038; 1 &#038; z\\\\<br \/>\n0 &#038; 1 &#038; t\\end{vmatrix}=-x-z+3t=0\\]<br \/>\n\\[\\begin{vmatrix}1 &#038; -1 &#038; y\\\\<br \/>\n-1 &#038; 1 &#038; z\\\\<br \/>\n0 &#038; 1 &#038; t\\end{vmatrix}=-y-z=0\\]<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\">(%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\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/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\">a<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/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\">transpose<\/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_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/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 &amp; 2 &amp; x\\\\1 &amp; -1 &amp; y\\\\-1 &amp; 1 &amp; z\\\\0 &amp; 1 &amp; t\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/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\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 2 &amp; x\\\\0 &amp; -3 &amp; y-x\\\\0 &amp; 0 &amp; z+y\\\\0 &amp; 1 &amp; t\\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\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 2 &amp; x\\\\0 &amp; 1 &amp; t\\\\0 &amp; 0 &amp; z+y\\\\0 &amp; -3 &amp; y-x\\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\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 2 &amp; x\\\\0 &amp; 1 &amp; t\\\\0 &amp; 0 &amp; z+y\\\\0 &amp; 0 &amp; y-x+3 t\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<p>Este resultado nos proporciona un m\u00e9todo muy pr\u00e1ctico para encontrar las ecuaciones ecuaciones impl\u00edcitas:<\/p>\n<ul>\n<li>Consideremos la matriz \\(A=\\begin{bmatrix}v_{11}&#038; v_{21}&#038;\\cdots &#038;v_{r1} &#038;x_1-p_1\\\\ v_{12}&#038; v_{22}&#038;\\cdots &#038;v_{r2} &#038;x_2-p_2\\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots &#038;\\vdots\\\\ v_{1n}&#038; v_{2n}&#038;\\cdots &#038;v_{rn} &#038;x_n-p_n\\end{bmatrix}\\), donde \\(rang(A)=r\\)<\/li>\n<li>Tomemos un menor de orden \\(r\\) distinto de cero; por ejemplo \\(\\begin{vmatrix}v_{11}&#038; v_{21}&#038;\\cdots &#038;v_{r1}\\\\ v_{12}&#038; v_{22}&#038;\\cdots &#038;v_{r2} \\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots \\\\ v_{1r}&#038; v_{2r}&#038;\\cdots &#038;v_{rn} \\end{vmatrix}\\neq 0\\)  <\/li>\n<li>Realicemos operaciones elementales a la matriz \\(A\\) de forma que mantengamos el menor anterior y hagamos cero el resto de filas. Esto nos llevar\u00e1 a una matriz semejante: \\[\\begin{bmatrix}v_{11}&#038; v_{21}&#038;\\cdots &#038;v_{r1} &#038;x_1-p_1\\\\ v_{12}&#038; v_{22}&#038;\\cdots &#038;v_{r2} &#038;x_2-p_2\\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots &#038;\\vdots\\\\ v_{1n}&#038; v_{2n}&#038;\\cdots &#038;v_{rn} &#038;x_n-p_n\\end{bmatrix}\\sim \\begin{bmatrix}v_{11}&#038; v_{21}&#038;\\cdots &#038;v_{r1} &#038;x_1-p_1\\\\ v_{12}&#038; v_{22}&#038;\\cdots &#038;v_{r2} &#038;x_2-p_2\\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots &#038;\\vdots\\\\<br \/>\nv_{1r}&#038; v_{2r}&#038;\\cdots &#038;v_{rr} &#038;x_r-p_r \\\\ 0&#038; 0&#038;\\cdots &#038;0&#038;f_1(x_1,\\ldots,x_n) \\\\<br \/>\n0&#038; 0&#038;\\cdots &#038;0&#038;f_2(x_1,\\ldots,x_n) \\\\ \\vdots&#038;\\vdots&#038;\\cdots &#038; \\vdots &#038;\\vdots\\\\<br \/>\n0&#038; 0&#038;\\cdots &#038;0&#038;f_{n-r}(x_1,\\ldots,x_n) \\end{bmatrix}\\]<\/li>\n<li>Las ecuaciones \\(f_i(x_1,\\ldots,x_n)=0\\), \\(i\\in\\{1,\\ldots,n-r\\}\\), definen las ecuaciones impl\u00edcitas de la variedad af\u00edn.<\/li>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas de la variedad lineal \\(\\textbf{S}=\\textbf{Gen}\\{1+X\\), \\(-X^2+2X^3\\}\\subseteq \\mathbb{R}_3[X]\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1c() {\n  var htmlShow1c = document.getElementById(\"html-show1c\");\n  if (htmlShow1c.style.display === \"none\") {\n    htmlShow1c.style.display = \"block\";\n  } else {\n    htmlShow1c.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1c\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Ecuaciones Impl\u00edcitas. Ej.4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/QD70eTDQAC4?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><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\">(%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\">p<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">coeff<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/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_number\">1<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">X<\/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_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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 &amp; 0 &amp; x\\\\1 &amp; 0 &amp; y\\\\0 &amp; -1 &amp; z\\\\0 &amp; 2 &amp; t\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/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_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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; x\\\\0 &amp; 0 &amp; y-x\\\\0 &amp; -1 &amp; z\\\\0 &amp; 2 &amp; t\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; x\\\\0 &amp; 0 &amp; y-x\\\\0 &amp; -1 &amp; z\\\\0 &amp; 0 &amp; 2 z+t\\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\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; x\\\\0 &amp; -1 &amp; z\\\\0 &amp; 0 &amp; y-x\\\\0 &amp; 0 &amp; 2 z+t\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas de la variedad af\u00edn dada por el punto P(1,0,-1,1,0) y el subespacio generado por los vectores \\(\\textbf{Gen}\\{(1,1,2,1,0),(-1,0,0,1,1)\\}\\)<\/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\/Ejer_variedad_afin01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas de la variedad af\u00edn \\(S=\\{(1,-2,3,1)+\\textbf{Gen}\\{(1,0,1,0)\\), \\((0,1,0,1)\\), \\((0,0,1,0)\\}\\}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1a() {\n  var htmlShow1a = document.getElementById(\"html-show1a\");\n  if (htmlShow1a.style.display === \"none\") {\n    htmlShow1a.style.display = \"block\";\n  } else {\n    htmlShow1a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Ecuaciones Impl\u00edcitas. Ej.5 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/zYo0TzTPohg?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><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\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <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; x-1\\\\0 &amp; 1 &amp; 0 &amp; y+2\\\\1 &amp; 0 &amp; 1 &amp; z-3\\\\0 &amp; 1 &amp; 0 &amp; t-1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/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_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x-1\\\\0 &amp; 1 &amp; 0 &amp; y+2\\\\0 &amp; 0 &amp; 1 &amp; z-x-2\\\\0 &amp; 1 &amp; 0 &amp; t-1\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x-1\\\\0 &amp; 1 &amp; 0 &amp; y+2\\\\0 &amp; 0 &amp; 1 &amp; z-x-2\\\\0 &amp; 0 &amp; 0 &amp; -y+t-3\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determina las ecuaciones impl\u00edcitas de la variedad af\u00edn dada por el punto P(-1,2,3,1,0) y el subespacio generado por los vectores \\(\\textbf{Gen}\\{(1,0,1,0,-1)\\), \\((0,1,0,1,1)\\), \\((0,0,-1,0,1)\\}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1b() {\n  var htmlShow1b = document.getElementById(\"html-show1b\");\n  if (htmlShow1b.style.display === \"none\") {\n    htmlShow1b.style.display = \"block\";\n  } else {\n    htmlShow1b.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1b\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Ecuaciones Impl\u00edcitas. Ej.6 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/jR5Yd734szo?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><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\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <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; x+1\\\\0 &amp; 1 &amp; 0 &amp; y-2\\\\1 &amp; 0 &amp; -1 &amp; z-3\\\\0 &amp; 1 &amp; 0 &amp; t-1\\\\-1 &amp; 1 &amp; 1 &amp; u\\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\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/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_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x+1\\\\0 &amp; 1 &amp; 0 &amp; y-2\\\\0 &amp; 0 &amp; -1 &amp; z-x-4\\\\0 &amp; 0 &amp; 0 &amp; -y+t+1\\\\-1 &amp; 1 &amp; 1 &amp; u\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; x+1\\\\0 &amp; 1 &amp; 0 &amp; y-2\\\\0 &amp; 0 &amp; -1 &amp; z-x-4\\\\0 &amp; 0 &amp; 0 &amp; -y+t+1\\\\0 &amp; 0 &amp; 0 &amp; z-y+u-1\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Componente normal<\/h2>\n<p>Para algunos ejercicios utilizaremos la siguiente definici\u00f3n:<\/p>\n<blockquote><p><strong>Definici\u00f3n:<\/strong> Sea \\(\\lambda_1x_1+\\lambda_2x_2+\\ldots+\\lambda_nx_n+\\lambda=0\\) una ecuaci\u00f3n impl\u00edcita de una variedad af\u00edn. Llamaremos <strong>componente normal<\/strong> de la ecuaci\u00f3n al vector \\((\\lambda_1,\\lambda_2,\\ldots,\\lambda_n)\\in\\mathbb{R}^n\\)<\/p>\n<p>Diremos <strong>componente normal unitaria<\/strong> a dicho vector multiplicado por el escalar: \\[\\frac{1}{\\sqrt{\\lambda_1^2+\\ldots+\\lambda_n^2}}(\\lambda_1,\\lambda_2,\\ldots,\\lambda_n)\\]<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1nto es el producto escalar de [1,2,3,4] por la componente normal unitaria de la ecuaci\u00f3n impl\u00edcita de la variedad af\u00edn \\(S=\\{(-1,2,3,1)+\\textbf{Gen}\\{(1,0,1,0)\\), \\((0,1,0,1)\\), \\((0,0,1,0)\\}\\}\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1e() {\n  var htmlShow1e = document.getElementById(\"html-show1e\");\n  if (htmlShow1e.style.display === \"none\") {\n    htmlShow1e.style.display = \"block\";\n  } else {\n    htmlShow1e.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1e()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1e\" style=\"display: none;\">\nEn el ejercicio anterior vimos que \\(S=\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ t-y=3\\}\\), luego la componente normal unitaria ser\u00e1: \\[\\left[ 0\\operatorname{,}-\\frac{1}{\\sqrt{2}}\\operatorname{,}0\\operatorname{,}\\frac{1}{\\sqrt{2}}\\right] \\]<br \/>\nAs\u00ed \\[[1,2,3,4].\\left[ 0\\operatorname{,}-\\frac{1}{\\sqrt{2}}\\operatorname{,}0\\operatorname{,}\\frac{1}{\\sqrt{2}}\\right]=\\sqrt{2}. \\]<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\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">nu<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">nu<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/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>\\[\\left[ 0,-\\frac{1}{\\sqrt{2}},0,\\frac{1}{\\sqrt{2}}\\right] \\]<\/p>\n<p>\\[\\sqrt{2}\\]<\/p>\n<p>\\[1.414213562373095\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Vector de coeficientes<\/h2>\n<p>Para algunos ejercicios utilizaremos la siguiente definici\u00f3n:<\/p>\n<blockquote><p><strong>Definici\u00f3n:<\/strong> Sea \\(\\lambda_1x_1+\\lambda_2x_2+\\ldots+\\lambda_nx_n+\\lambda=0\\) una ecuaci\u00f3n impl\u00edcita de una variedad af\u00edn. Llamaremos <strong>vector de coeficientes<\/strong> de la ecuaci\u00f3n al vector \\((\\lambda_1,\\lambda_2,\\ldots,\\lambda_n,\\lambda)\\in\\mathbb{R}^{n+1}\\)<\/p>\n<p>Diremos <strong>vector de coeficientes unitario<\/strong> a dicho vector multiplicado por el escalar: \\[\\frac{1}{\\sqrt{\\lambda_1^2+\\ldots+\\lambda_n^2+\\lambda^2}}(\\lambda_1,\\lambda_2,\\ldots,\\lambda_n,\\lambda)\\]<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar si [4,7,-3,3,-6,-4] pertenece al subespacio generado por los <em>vectores de coeficientes<\/em> de las ecuaciones impl\u00edcitas de la variedad af\u00edn \\(S=\\{[-1,2,3,1,0]+\\textbf{Gen}\\{([2,1,1,0,2]\\), \\([-1,1,0,-1,0]\\), \\([0,0,-1,1,1]\\}\\}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1e7() {\n  var htmlShow1e7 = document.getElementById(\"html-show1e7\");\n  if (htmlShow1e7.style.display === \"none\") {\n    htmlShow1e7.style.display = \"block\";\n  } else {\n    htmlShow1e7.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1e7()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1e7\" style=\"display: none;\">\nPlanteemos la matriz de donde obtendremos las ecuaciones impl\u00edcitas de la variedad: \\[\\begin{bmatrix}2 &#038; -1 &#038; 0 &#038; {x_1}+1\\\\<br \/>\n1 &#038; 1 &#038; 0 &#038; {x_2}-2\\\\ 1 &#038; 0 &#038; -1 &#038; {x_3}-3\\\\ 0 &#038; -1 &#038; 1 &#038; {x_4}-1\\\\ 2 &#038; 0 &#038; 1 &#038; {x_5}\\end{bmatrix}\\]<\/p>\n<p>Como el rango del subespacio director es 3, nos ser\u00e1 suficiente con un menor de orden 3 distinto de cero, por ejemplo: \\[\\begin{bmatrix}2 &#038; -1 &#038; 0 &#038; {x_1}+1\\\\ 1 &#038; 1 &#038; 0 &#038; {x_2}-2\\\\ 1 &#038; 0 &#038; -1 &#038; {x_3}-3\\end{bmatrix}\\]<br \/>\nAhora, hacemos los dos menores de orden cuatro restantes con las filas que hemos omitido, y calculamos su determinante: \\[\\begin{vmatrix}2 &#038; -1 &#038; 0 &#038; {x_1}+1 \\\\ 1 &#038; 1 &#038; 0 &#038; {x_2}-2\\\\ 1 &#038; 0 &#038; -1 &#038; {x_3}-3\\\\ 0 &#038; -1 &#038; 1 &#038; {x_4}-1\\end{vmatrix}=-3 {x_4}-3 {x_3}-{x_2}+2 {x_1}+16=0\\] y \\[\\begin{vmatrix}2 &#038; -1 &#038; 0 &#038; {x_1}+1\\\\ 1 &#038; 1 &#038; 0 &#038; {x_2}-2\\\\ 1 &#038; 0 &#038; -1 &#038; {x_3}-3\\\\ 2 &#038; 0 &#038; 1 &#038; {x_5}\\end{vmatrix}=-3 {x_3}+3 {x_2}+3 {x_1}-3 {x_5}+6=0\\]<\/p>\n<p>Solo nos resta disponer en una matriz los vectores de coeficientes de cada ecuaci\u00f3n junto con el vector [4,7,-3,3,-6,-4] y calcular su rango:<br \/>\n\\[\\mathbf{rang}\\begin{bmatrix}2 &#038; -1 &#038; -3 &#038; -3 &#038; 0 &#038; 16\\\\ 3 &#038; 3 &#038; -3 &#038; 0 &#038; -3 &#038; 6\\\\ 4 &#038; 7 &#038; -3 &#038; 3 &#038; -6 &#038; -4\\end{bmatrix}=2\\]<br \/>\nEsto nos confirma que [4,7,-3,3,-6,-4] pertenece al subespacio generado por los <em>vectores de coeficientes<\/em> de las ecuaciones impl\u00edcitas de la variedad af\u00edn \\(S\\).<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\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <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\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_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_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <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}2 &amp; -1 &amp; 0 &amp; x+1\\\\1 &amp; 1 &amp; 0 &amp; y-2\\\\1 &amp; 0 &amp; -1 &amp; z-3\\\\0 &amp; -1 &amp; 1 &amp; t-1\\\\2 &amp; 0 &amp; 1 &amp; u\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 1 &amp; 0 &amp; y-2\\\\2 &amp; -1 &amp; 0 &amp; x+1\\\\1 &amp; 0 &amp; -1 &amp; z-3\\\\0 &amp; -1 &amp; 1 &amp; t-1\\\\2 &amp; 0 &amp; 1 &amp; u\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\">thru<\/span><span class=\"code_number\">5<\/span><span class=\"code_function\">do<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/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_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 1 &amp; 0 &amp; y-2\\\\0 &amp; -3 &amp; 0 &amp; -2 \\left( y-2\\right) +x+1\\\\0 &amp; -1 &amp; -1 &amp; z-y-1\\\\0 &amp; -1 &amp; 1 &amp; t-1\\\\0 &amp; -2 &amp; 1 &amp; u-2 \\left( y-2\\right) \\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">3<\/span><span class=\"code_function\">thru<\/span><span class=\"code_number\">5<\/span><span class=\"code_function\">do<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 1 &amp; 0 &amp; y-2\\\\0 &amp; -3 &amp; 0 &amp; -2 \\left( y-2\\right) +x+1\\\\0 &amp; 0 &amp; -1 &amp; z-y-\\frac{-2 \\left( y-2\\right) +x+1}{3}-1\\\\0 &amp; 0 &amp; 1 &amp; -\\frac{-2 \\left( y-2\\right) +x+1}{3}+t-1\\\\0 &amp; 0 &amp; 1 &amp; -2 \\left( y-2\\right) -\\frac{2 \\left( -2 \\left( y-2\\right) +x+1\\right) }{3}+u\\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\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">4<\/span><span class=\"code_function\">thru<\/span><span class=\"code_number\">5<\/span><span class=\"code_function\">do<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 1 &amp; 0 &amp; y-2\\\\0 &amp; -3 &amp; 0 &amp; -2 \\left( y-2\\right) +x+1\\\\0 &amp; 0 &amp; -1 &amp; z-y-\\frac{-2 \\left( y-2\\right) +x+1}{3}-1\\\\0 &amp; 0 &amp; 0 &amp; z-y-\\frac{2 \\left( -2 \\left( y-2\\right) +x+1\\right) }{3}+t-2\\\\0 &amp; 0 &amp; 0 &amp; z-y-x+u-2\\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\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/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\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\frac{3 z+y-2 x+3 t-16}{3},z-y-x+u-2\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <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_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">16<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[2\\]<\/p>\n<\/div>\n<hr \/>\n<p>&nbsp;<\/p>\n<h3>De impl\u00edcitas a param\u00e9tricas<\/h3>\n<p>Pasar de las ecuaciones impl\u00edcitas a las param\u00e9tricas ser\u00e1 resolver un sistema.<\/p>\n<p>La introducci\u00f3n de las variedades nos lleva a resolver sistemas de ecuaciones lineales. Para resolverlas utilizamos las matrices. As\u00ed todo sistema de ecuaciones lineales lo podemos plantear como un sistema matricial de la forma <strong>Ax<\/strong>=<strong>b<\/strong>, donde <strong>A<\/strong> es la matriz de coeficientes del sistema, <strong>x<\/strong> es la matriz columna de inc\u00f3gnitas y <strong>b<\/strong> es la matriz columna de t\u00e9rminos independientes:<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"alignnone aligncenter\" alt=\"\" src=\"http:\/\/upload.wikimedia.org\/math\/d\/3\/d\/d3d71806a01f5a3490b7d038c1650f43.png\" width=\"339\" height=\"109\" \/><\/p>\n<p>Para tratarlos mejor, podemos intentar transformarlos en sistemas escalonados, que es resultado de transformar la matriz ampliada <strong>[A b]<\/strong>, mediante operaciones elementales de fila, en una matriz escalonada. Este es el m\u00e9todo que conocemos como <a href=\"http:\/\/es.wikipedia.org\/wiki\/Eliminaci%C3%B3n_de_Gauss-Jordan\" target=\"_blank\" rel=\"noopener noreferrer\">m\u00e9todo de Gauss<\/a>. <\/p>\n<p>Los sistemas de ecuaciones m\u00e1s sencillos resultan aquellos que podemos emplear la<a href=\"http:\/\/es.wikipedia.org\/wiki\/Regla_de_Cramer\" target=\"_blank\" rel=\"noopener noreferrer\"> regla de Cramer<\/a>.<\/p>\n<p>La importancia de <a href=\"http:\/\/es.wikipedia.org\/wiki\/Teorema_de_Rouch%C3%A9%E2%80%93Frobenius\" target=\"_blank\" rel=\"noopener noreferrer\">Teorema de Rouch\u00e9-Frobenius <\/a>estriba en que determina cuando un sistema tiene soluci\u00f3n o no. Este resultado, junto con el anterior, nos permite resolver con facilidad los sistemas de ecuaciones como los ejercicios que hemos realizado.<\/p>\n<p>&nbsp;<\/p>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong> La distancia al origen de la recta r \u2261 3x &#8211; 4y &#8211; 25 = 0 es <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>25<\/li>\n<li>5<\/li>\n<li>3\/4<\/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>Ampliamos las definiciones de variedades lineales que, en muchos casos, las equiparamos con los subespacios vectoriales, aunque no tienen por qu\u00e9 serlos, a \\(\\mathbb{R}^n\\) Las variedades lineales nos dan pie para definir&#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":[20],"class_list":["post-461","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/461","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=461"}],"version-history":[{"count":5,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/461\/revisions"}],"predecessor-version":[{"id":474,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/461\/revisions\/474"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}