{"id":563,"date":"2025-11-26T08:19:40","date_gmt":"2025-11-26T07:19:40","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=563"},"modified":"2025-11-27T11:27:28","modified_gmt":"2025-11-27T10:27:28","slug":"alg-proyeccion-ortogonal","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=563","title":{"rendered":"ALG: Proyecci\u00f3n ortogonal"},"content":{"rendered":"<p>El pasado d\u00eda ve\u00edamos que cuando \\(S\\) era un subespacio vectorial entonces \\[\\mathcal{E}=S\\oplus S^{\\bot}\\]<\/p>\n<p>Esto implica que para todo vector \\(\\vec{v}\\in \\mathcal{E}\\) existir\u00e1n dos \u00fanicos vectores \\(\\vec{u}\\in S\\) y \\(\\vec{w}\\in S^{\\bot}\\), tales que \\[\\vec{v}=\\vec{u}+\\vec{w}.\\]<\/p>\n<p>Estos vectores \\(\\vec{u}\\) o \\(\\vec{w}\\) son lo que llamamos proyecciones ortogonales de \\(\\vec{v}\\) sobre \\(S\\) o \\(S^{\\bot}\\) respectivamente.<\/p>\n<p>La definici\u00f3n cl\u00e1sica nos dice que si \\(S\\subset \\mathcal{E}\\), un subespacio vectorial de un espacio eucl\u00eddeo, para nuestros casos finitamente generado, llamamos proyecci\u00f3n ortogonal del vector \\(\\vec{v}\\) sobre el subespacio \\(S\\), al \u00fanico vector \\(\\vec{u}\\in S\\) talque \\(\\vec{v}-\\vec{u}\\in S^{\\bot}\\).<\/p>\n<p>A la aplicaci\u00f3n \\(\\mathbf{proy}_S:\\mathcal{E}\\to S\\) que a cada vector de \\(\\mathcal{E}\\) le hace corresponder su proyecci\u00f3n ortogonal sobre \\(S\\), se le denomina del mismo modo: proyecci\u00f3n ortogonal.<\/p>\n<p>Veamos un m\u00e9todo para calcular la proyecci\u00f3n ortogonal. Primero empezamos con la proyecci\u00f3n sobre un vector. Si \\(S=&lt;\\vec{s}&gt;\\); es decir, es una recta, entonces \\[\\mathbf{proy}_\\vec{s}(\\vec{v})=\\frac{\\vec{v}\\bullet\\vec{s}}{\\parallel\\vec{s}\\parallel^2}\\vec{s}.\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Hallar la proyecci\u00f3n de \\(A:\\begin{bmatrix}1 &#038; 2\\\\ -1 &#038; 3\\end{bmatrix}\\) sobre \\(B:\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\) <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1q2() {\n  var htmlShow1q2 = document.getElementById(\"html-show1q2\");\n  if (htmlShow1q2.style.display === \"none\") {\n    htmlShow1q2.style.display = \"block\";\n  } else {\n    htmlShow1q2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1q2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1q2\" style=\"display: none;\">\n\\[\\begin{split}<br \/>\n\\mathbf{proy}_BA&#038;=\\frac{A\\bullet B}{B\\bullet B}B=<br \/>\n\\frac{\\mathbf{tr}\\left(\\begin{bmatrix}1 &#038; 2\\\\ -1 &#038; 3\\end{bmatrix}^t\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\right)}<br \/>\n     {\\mathbf{tr}\\left(\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}^t\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\right)}<br \/>\n\\ \\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\\\<br \/>\n&#038;=\\frac{\\mathbf{tr}\\left(\\begin{bmatrix}1 &#038; 3\\\\ 2 &#038; -4\\end{bmatrix}\\right)}<br \/>\n     {\\mathbf{tr}\\left(\\begin{bmatrix}1 &#038; 1\\\\ 1 &#038; 5\\end{bmatrix}\\right)}<br \/>\n\\ \\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\\\<br \/>\n&#038;=\\frac{-3}{6}\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}\\\\<br \/>\n&#038;=\\frac{-1}{2}\\begin{bmatrix}1 &#038; 1\\\\ 0 &#038; -2\\end{bmatrix}<br \/>\n\\end{split}\\]\n<\/div>\n<hr \/>\n<p>Como hemos dicho, para todo vector \\(\\vec{v}\\in \\mathcal{E}\\) existir\u00e1n dos \u00fanicos vectores \\(\\vec{u}\\in S\\) y \\(\\vec{w}\\in S^{\\bot}\\), tales que \\[\\vec{v}=\\vec{u}+\\vec{w}.\\]<\/p>\n<p>Luego determinando la proyecci\u00f3n de un vector sobre un subespacio puedes obtener de manera indirecta la proyecci\u00f3n sobre su ortogonal.<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Hallar \\(u:[2,3,2]\\) como suma directa del subespacios vectorial \\(\\mathbf{Gen}\\{[1,2,-6]\\}\\) y su ortogonal.<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv13a2() {\n  var htmlShow13a2 = document.getElementById(\"html-show13a2\");\n  if (htmlShow13a2.style.display === \"none\") {\n    htmlShow13a2.style.display = \"block\";\n  } else {\n    htmlShow13a2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv13a2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show13a2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Suma Directa Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/_ApbvnO3heI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>Extenderlo a cualquier subespacio es sencillo, solo necesitamos una base ortogonal del subespacio: Sea \\( \\{\\vec{u}_1,\\vec{u}_1,\\ldots,\\vec{u}_m\\}\\) una base ortogonal de \\(S\\), entonces<br \/>\n\\[proy_S(\\vec{v})=\\sum_{i=1}^m\\frac{\\vec{v}\\bullet\\vec{u}_i}{\\parallel\\vec{u}_i\\parallel^2}\\vec{u}_i.\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(S:\\left\\{\\begin{bmatrix}3a+2b &#038; -2a-b\\\\ b &#038; a\\end{bmatrix}\\in\\mathcal{M}_2(\\mathbb{R}) \\right\\}\\). \u00bfCu\u00e1l es la \\(\\left\\|\\textbf{proy}_S\\left(\\begin{bmatrix}-1 &#038; 0\\\\ 2 &#038; 1\\end{bmatrix}\\right)\\right\\|\\) <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1q3() {\n  var htmlShow1q3 = document.getElementById(\"html-show1q3\");\n  if (htmlShow1q3.style.display === \"none\") {\n    htmlShow1q3.style.display = \"block\";\n  } else {\n    htmlShow1q3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1q3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1q3\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Proyecci\u00f3n Octogonal Ejercicio 4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/r8BTACnZkWI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(S:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+y-z=0,\\ x-y+3t=0\\}\\). \u00bfCu\u00e1l es la \\(\\|\\textbf{proy}_S([-1,0,2,1])\\|\\)?\n <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1q4() {\n  var htmlShow1q4 = document.getElementById(\"html-show1q4\");\n  if (htmlShow1q4.style.display === \"none\") {\n    htmlShow1q4.style.display = \"block\";\n  } else {\n    htmlShow1q4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1q4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1q4\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Proyecci\u00f3n Octogonal Ejercicio 3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/a0Brp8GggNw?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>Si adem\u00e1s la base es ortonormal la expresi\u00f3n se reduce mucho:<br \/>\n\\[proy_S(\\vec{v})=(\\vec{v}\\bullet\\vec{u}_i)\\vec{u}_1+(\\vec{v}\\bullet\\vec{u}_2)\\vec{u}_2+\\ldots+(\\vec{v}\\bullet\\vec{u}_m)\\vec{u}_m.\\]<\/p>\n<p>El prop\u00f3sito es determinar dado un subespacio vectorial \\(S\\subset\\mathbb{R}^n\\) y un vector, o punto, \\(\\vec{v}\\in\\mathbb{R}^n\\), minimizar la distancia de \\(\\vec{v}\\) a cualquier \\(\\vec{s}\\in S\\). Para conseguirlo utilizamos el siguiente resultado:<\/p>\n<blockquote>\n<p><strong>Teorema:<\/strong> Sea \\(S\\subset\\mathbb{R}^n\\) un sube.v., \\(\\vec{v}\\in\\mathbb{R}^n\\) y \\(\\vec{s}\\in S\\), son equivalentes<\/p>\n<ol>\n<li>\\(\\vec{s}\\in S\\) es la proyecci\u00f3n ortogonal de \\(\\vec{v}\\) sobre \\(S\\), \\(proy_S(\\vec{v})\\); es decir, \\(\\vec{v}-\\vec{s}\\in S^{\\bot}\\)<\/li>\n<li>\\(\\vec{s}\\in S\\) es la mejor aproximaci\u00f3n de \\(\\vec{v}\\) sobre \\(S\\); es decir,\\(\\parallel \\vec{v}-\\vec{s}\\parallel\\leq \\parallel \\vec{v}-\\vec{w}\\parallel\\,\\forall \\vec{w}\\in S\\)<\/li>\n<\/ol>\n<\/blockquote>\n<p>En ejemplo lo pod\u00e9is ver el la deducci\u00f3n de la distancia entre un punto \\(P(x_0,y_0)\\) y la recta \\(r:ax+by+c=0\\) que viene dada por la f\u00f3rmula \\[d(P,r)=\\frac{ax_0+by_0+c}{\\sqrt{a^2+b^2}}\\]<br \/>\nEn este enlace est\u00e1 la demostraci\u00f3n <a href=\"http:\/\/youtu.be\/JaDJqaOYpJU\">Proyecci\u00f3n Ortogonal. Ej.1<\/a><\/p>\n<h3>M\u00ednimos cuadrados<\/h3>\n<p>Hemos visto c\u00f3mo solucionar sistemas de ecuaciones lineales. Sin embargo, hay en ocasiones que los sistemas no tiene soluci\u00f3n. En estos casos podemos buscar el punto m\u00e1s cercano a la soluci\u00f3n. Recordemos que todo sistema podemos plantearlo en su forma matricial como<br \/>\n\\[    A\\ x=\\ \\textbf{b},\\]<br \/>\ndonde \\(A\\in\\mathcal{M}_{m\\times n}(\\mathbb{R})\\), \\(x\\in\\mathcal{M}_{n\\times 1}(\\mathbb{R})\\) y \\(\\textbf{b}\\in\\mathcal{M}_{m\\times 1}(\\mathbb{R})\\). <\/p>\n<p>Cuando se necesita una soluci\u00f3n pero no hay ninguna, lo mejor que puede hacerse es encontrar una soluci\u00f3n \\(x\\) que deje a \\(A\\ x\\) tan cercana a \\(\\textbf{b}\\) como sea posible.<\/p>\n<blockquote>\n<p>\nSea \\( A\\in\\mathcal{M}_{m\\times n}(\\mathbb{R})\\), \\( x^t\\in\\mathbb{R}^n\\) y \\(\\textbf{b}^t\\in\\mathbb{R}^m\\), llamamos soluci\u00f3n por m\u00ednimos cuadrados de la ecuaci\u00f3n, a una aproximaci\u00f3n \\( \\hat{x}^t\\in\\mathbb{R}^n\\), tal que<br \/>\n\\[\\parallel\\textbf{b}-A\\ \\hat{x}\\parallel\\leq \\parallel \\textbf{b}-A\\ x\\parallel\\ \\forall\\ x^t\\in\\mathbb{R}^n.\\]\n<\/p>\n<\/blockquote>\n<p>El conjunto de soluciones por m\u00ednimos cuadrados de \\( A\\ x=\\ \\textbf{b}\\) coincide con el conjunto no vac\u00edo de soluciones de<br \/>\n\\[    A^t A\\ x=\\ A^t\\textbf{b},\\]<\/p>\n<p>De la propiedad anterior se deduce un resultado concluyente:<\/p>\n<blockquote>\n<p>\nSi las columnas de \\(A\\) son linealmente independientes, entonces \\(A^t A\\) es invertible y la ecuaci\u00f3n \\(A\\ x=\\ \\textbf{b}\\) tiene solamente una soluci\u00f3n por m\u00ednimos cuadrados dada por<br \/>\n\\[ \\hat{x}=\\ (A^t A)^{-1} A^t\\textbf{b}.\\]\n<\/p>\n<\/blockquote>\n<p>Esta forma de calcular la soluci\u00f3n por m\u00ednimos cuadrados ser\u00eda equivalente a considerar \\(\\bar{A}\\) el subespacio vectorial de \\(\\mathbb{R}^m\\) generado por los vectores columna de \\(A\\) y determinar<br \/>\n\\[   \\hat{\\textbf{b}}=proy_{\\bar{A}}(\\textbf{b}).\\]<br \/>\nEntonces<br \/>\n\\[ A\\ \\hat{x}=\\ \\hat{\\textbf{b}}.\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Cu\u00e1l es el error de una soluci\u00f3n por m\u00ednimos cuadrados del sistema incompatible<br \/>\n\\[\\begin{array}{r}  -x +2y=4, \\\\     2x-3y=1, \\\\    -x+3y=2.\\end{array}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1q2e2() {\n  var htmlShow1q2e2 = document.getElementById(\"html-show1q2e2\");\n  if (htmlShow1q2e2.style.display === \"none\") {\n    htmlShow1q2e2.style.display = \"block\";\n  } else {\n    htmlShow1q2e2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1q2e2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1q2e2\" style=\"display: none;\">\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_minimos_cuadrados.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>&nbsp;<\/p>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong>Sea \\(\\pi:\\{(x,y,z,t)\\in\\mathbb{R}^4;\\ 2x+3y-z=0,\\ y+2z-t=0\\) un plano en \\(\\mathbb{R}^4\\). \u00bfCu\u00e1l de los vectores a:[8,13,-2,-1], b:[8,-13,2,-1] y c:[-8,13,-2,1], pertenece a su ortogonal? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>a<\/li>\n<li>b<\/li>\n<li>c<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>A.)<\/strong><\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El ortogonal, \\(\\pi^\\bot\\), est\u00e1 formado por los vectores ortogonales a \\(\\pi\\), y para ello necesitamos los vectores de una base del subespacio:<\/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\">(%i1) <\/span><\/td>\n<td><span class=\"input\"><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/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\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/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_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ x=-\\frac{3 {\\mathrm{\\% r1}}-7 {\\mathrm{\\% r2}}}{2}\\operatorname{,}y={\\mathrm{\\% r1}}-2 {\\mathrm{\\% r2}}\\operatorname{,}z={\\mathrm{\\% r2}}\\operatorname{,}t={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esto nos da los vectores:<\/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\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_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_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%o1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r1<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r2<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/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\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_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_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%o1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r1<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r2<\/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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ -\\frac{3}{2}\\operatorname{,}1\\operatorname{,}0\\operatorname{,}1\\right] \\]<\/p>\n<p>\\[\\left[ \\frac{7}{2}\\operatorname{,}-2\\operatorname{,}1\\operatorname{,}0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Como da los mismo utilizar cualquiera que sea combinaci\u00f3n de ellos, podemos eliminar las fracciones:<\/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\">(%i5) <\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ -3\\operatorname{,}2\\operatorname{,}0\\operatorname{,}2\\right] \\]<\/p>\n<p>\\[\\left[ 7\\operatorname{,}-4\\operatorname{,}2\\operatorname{,}0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El ortogonal, \\(\\pi^\\bot\\), est\u00e1 formado por los vectores cuyo producto escalar sea cero:<\/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\">(%i7) <\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">v<\/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_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/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_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[2 y-3 x+2 t=0\\]<\/p>\n<p>\\[2 z-4 y+7 x=0\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Estas ecuaciones definen el producto escalar. <br \/>Si queremos dar las ecuaciones param\u00e9tricas resolvemos el sistema.<\/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><span class=\"input\"><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">[<\/span><span class=\"code_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_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/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_number\">0<\/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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ x={\\mathrm{\\% r3}}\\operatorname{,}y=\\frac{2 {\\mathrm{\\% r4}}+7 {\\mathrm{\\% r3}}}{4}\\operatorname{,}z={\\mathrm{\\% r4}}\\operatorname{,}t=-\\frac{2 {\\mathrm{\\% r4}}+{\\mathrm{\\% r3}}}{4}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esto nos da los vectores, simplificando y quitando la fracci\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\">(%i10)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_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_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%o8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r3<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r4<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_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_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%o8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r3<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r4<\/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\">\u00b7<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ 4\\operatorname{,}7\\operatorname{,}0\\operatorname{,}-1\\right] \\]<\/p>\n<p>\\[\\left[ 0\\operatorname{,}1\\operatorname{,}2\\operatorname{,}-1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Los vectores a:[8,13,-2,-1], b:[8,-13,2,-1] y c:[-8,13,-2,1], pertenece al ortogonal si son combinaci\u00f3n lineal de los anteriores. Veamos el rango<\/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\">(%i16)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">13<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">13<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">13<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><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_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><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_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><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_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[2\\]<\/p>\n<p>\\[3\\]<\/p>\n<p>\\[3\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Esto nos dice que el vector a pertenece \\(\\pi^\\bot\\).<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>El pasado d\u00eda ve\u00edamos que cuando \\(S\\) era un subespacio vectorial entonces \\[\\mathcal{E}=S\\oplus S^{\\bot}\\] Esto implica que para todo vector \\(\\vec{v}\\in \\mathcal{E}\\) existir\u00e1n dos \u00fanicos vectores \\(\\vec{u}\\in S\\) y \\(\\vec{w}\\in S^{\\bot}\\), tales&#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-563","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/563","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=563"}],"version-history":[{"count":3,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/563\/revisions"}],"predecessor-version":[{"id":615,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/563\/revisions\/615"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}