{"id":2002,"date":"2017-07-04T21:45:21","date_gmt":"2017-07-05T02:45:21","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1003\/?p=2002"},"modified":"2017-07-04T21:45:21","modified_gmt":"2017-07-05T02:45:21","slug":"cl2-01-transformaciones-lineales","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/matg1049\/cl2-01-transformaciones-lineales\/","title":{"rendered":"cl3-01. Transformaciones Lineales"},"content":{"rendered":"

<\/code><\/p>\n

\n
Definici\u00f3n.<\/strong> Sean V<\/span> y W<\/span> espacios vectoriales cualesquiera. Una transformaci\u00f3n lineal T{:}\\ V\\rightarrow W<\/span> es una funci\u00f3n que asigna a cada vector v\\in V<\/span> un vector \u00fanico T(v)\\in W<\/span> y que satisface las siguientes condiciones:\n\n1)<\/span> \\forall\\ v_1, v_2\\in V{:}\\ T(v_1+v_2)=T(v_1)+T(v_2)<\/span>.\n2)<\/span> \\forall\\ v\\in V,\\ \\forall \\alpha\\in \\mathbb{R}{:}\\ T(\\alpha v)=\\alpha T(v)<\/span>.\n<\/pre>\n
Ejemplo.<\/strong> Sean V=\\mathbb{C}[a,b]<\/span> y W=\\mathbb{R}<\/span> espacios vectoriales. Demuestre que la transformaci\u00f3n T{:}\\ V\\rightarrow W<\/span> definida por T(f)=\\int_{a}^{b} f(x) dx<\/span> es lineal.<\/pre>\n
Soluci\u00f3n.<\/strong>
\n1)<\/span> \\forall\\ f, g\\in V{:}\\ T(f+g)=T(f)+T(g)<\/span>.<\/p>\n
\\small{T(f)=\\int_{a}^{b} f(x) dx \\qquad T(g)=\\int_{a}^{b} g(x) dx \\qquad (f+g)(x)=f(x)+g(x)}<\/span>\\begin{array}{rcl}T(f+g)&=&\\int_{a}^{b} (f+g)(x)dx\\\\T(f+g)&=&\\int_{a}^{b} [f(x)+g(x)]dx\\\\T(f)+T(g)&=&\\int_{a}^{b} f(x) dx + \\int_{a}^{b} g(x) dx\\end{array}<\/span><\/p>\n
2)<\/span> \\forall\\ f\\in V,\\ \\forall \\alpha\\in \\mathbb{R}{:}\\ T(\\alpha f)=\\alpha T(f)<\/span><\/p>\n
\\small{(\\alpha f)(x)=\\alpha f(x) \\qquad T(f)=\\int_{a}^{b} f(x) dx \\Longrightarrow T(\\alpha f)=\\int_{a}^{b} (\\alpha f)(x) dx}<\/span>\\small{\\int_{a}^{b} (\\alpha f)(x) dx = \\int_{a}^{b} \\alpha f(x) dx = \\alpha \\int_{a}^{b} f(x) dx = \\alpha T(f)}<\/span><\/p>\n
Por consiguiente, T{:}\\ V\\rightarrow W<\/span> es una transformaci\u00f3n lineal.<\/p>\n
\u00a0<\/p>\n
<\/p>\n
Teorema.<\/strong> Sea T{:}\\ V\\rightarrow W<\/span> una transformaci\u00f3n lineal entonces:\n\nLa imagen del cero vector de V<\/span> es el cero vector de W<\/span>.\nT(0_V)=0_W<\/span>\nLa imagen del inverso de V<\/span> es el inverso de V<\/span>.\nT(-v)=-T(v)<\/span>\nLa transformada de la combinaci\u00f3n lineal es la combinaci\u00f3n lineal de la transformada.\n\\small{T(\\alpha_1 V_1 + \\alpha_2 V_2 + ... +\\alpha_n V_n)=\\alpha_1 T(V_1) + \\alpha_2 T(V_2) + ... +\\alpha_n T(V_n)}<\/span><\/pre>\n
Ejemplo.<\/strong> Sea T{:}\\ \\wp_2 \\rightarrow \\mathbb{R^3}<\/span> una tranformaci\u00f3n lineal tal que:\\footnotesize{T(x+1)=\\left(\\begin{array}{r} 1 \\\\ 0\\\\ 1 \\end{array}\\right) \\quad T(x^2 -1)=\\left(\\begin{array}{r} 0 \\\\ 1\\\\ 1 \\end{array}\\right) \\quad T(2x^2+x-2)=\\left(\\begin{array}{r} 1 \\\\ 1\\\\ 0 \\end{array}\\right)}<\/span>Determine T(ax^2+bx+c)<\/span>.<\/pre>\n
Soluci\u00f3n.<\/strong><\/p>\n
T(ax^2+bx+c)=aT(x^2)+bT(x)+cT(1)<\/span>
\n\\left\\{ \\begin{array}{rcl}T(x)+T(1)&=&V_1 \\\\ T(x^2)-T(1)&=&V_2 \\\\ 2T(x^2)+T(x)-2T(1)&=&V_3 \\end{array}\\right.<\/span>
\n\\left(\\begin{array}{rrr|r} 0 & 1 & 1 & V_1\\\\ 1 & 0 & -1& V_2 \\\\ 2&1&-2&V_3 \\end{array}\\right) \\approx \\left(\\begin{array}{rrr|r} 1&0&-1&V_2 \\\\ 0&1&1&V_1 \\\\ 0&1&0&V_3 -2V_2 \\end{array}\\right) <\/span><\/p>\n\\begin{array}{rclcrcc} T(x)&=&V_3 -2V_2 &\\Longrightarrow& T(x)&=&{\\left(\\begin{array}{r} 1 \\\\ -1\\\\ 2 \\end{array}\\right)} \\\\ T(1)&=&V_1 - T(x) &\\Longrightarrow & T(1)&=&{\\left(\\begin{array}{r} 0 \\\\ 1\\\\ 3 \\end{array}\\right)} \\\\ T(x^2)&=&V_2 + T(1) &\\Longrightarrow & T(x^2)&=&{\\left(\\begin{array}{r} 0 \\\\ 2\\\\ 4 \\end{array}\\right)} \\end{array}<\/span>\nT(ax^2+bx+c) = a {\\left(\\begin{array}{r} 0 \\\\ 2\\\\ 4 \\end{array}\\right)} + b {\\left(\\begin{array}{r} 1 \\\\ -1\\\\ 2 \\end{array}\\right)} + c {\\left(\\begin{array}{r} 0 \\\\ 1\\\\ 3 \\end{array}\\right)}<\/span>\n
Por consiguiente, T(ax^2+bx+c) = {\\left(\\begin{array}{c} b \\\\ 2a-b+c\\\\ 4a-2b+3c \\end{array}\\right)} <\/span><\/p>\n
\u00a0<\/p>\n
<\/p>\n
Ejemplo.<\/strong> Se consideran los conjuntos de vectores R<\/span> y R'<\/span>, cuyas coordenadas se asocian a los puntos que forman los rombos ABCD<\/span> y A'B'C'D'<\/span> como se aprecia en la figura a continuaci\u00f3n:\n\n\"\"Si se define T{:}\\ \\mathbb{R^2}\\rightarrow \\mathbb{R^2}<\/span> como T(x,y)=(-y,2x)<\/span>, grafique R''=T(R')<\/span>; adem\u00e1s, verifique que T<\/span> es una transformaci\u00f3n lineal.\n<\/pre>\n
Soluci\u00f3n.<\/strong>
\n\\begin{array}{l} {A'=(x,y)=(4,-4)\\longrightarrow T(x,y)=(-y,2x)=(4,8)=A''}\\\\{B'=(x,y)=(6,-4)\\longrightarrow T(x,y)=(-y,2x)=(4,12)=B''}\\\\{C'=(x,y)=(2,-2)\\longrightarrow T(x,y)=(-y,2x)=(2,4)=C''}\\\\{D'=(x,y)=(4,-2)\\longrightarrow T(x,y)=(-y,2x)=(2,8)=D''}\\end{array}<\/span>Por consiguiente, el conjunto de vectores R''<\/span> tiene como coordenadas los puntos que conforman el rombo A''B''C''D''<\/span> como se aprecia en la figura a continuaci\u00f3n:<\/p>\n
\"\"<\/p>\n
Para verificar que T<\/span> es una transformaci\u00f3n lineal, por definici\u00f3n se debe satisfacer las siguientes condiciones:<\/p>\n
1)<\/span> \\forall\\ v_1, v_2\\in V{:}\\ T(v_1+v_2)=T(v_1)+T(v_2)<\/span>.<\/p>\n
Sean v_1=(x,y)<\/span> y v_2=(x',y')<\/span> dos vectores de \\mathbb{R^2}<\/span>.<\/br><\/p>\n\\begin{array}{rcl} T(v_1+v_2)&=&T((x,y)+(x',y')) \\\\ &=&T(x+x',y+y')\\\\ &=&(-(y+y'),2(x+x')) \\\\ &=&(-y-y',2x+2x') \\\\ &=&(-y,2x)+(-y',2x') \\\\ &=&T(v_1)+T(v_2)\\end{array}<\/span>\n
2)<\/span> \\forall\\ v\\in V,\\ \\forall \\alpha\\in \\mathbb{R}{:}\\ T(\\alpha v)=\\alpha T(v)<\/span>.<\/p>\n
Sean v=(x,y)<\/span> un vector de \\mathbb{R^2}<\/span> y \\alpha \\in \\mathbb{R}<\/span>.<\/br><\/p>\n\\begin{array}{rcl} T(\\alpha v)&=&T(\\alpha (x,y))\\\\&=&T(\\alpha x,\\alpha y)\\\\&=&(-\\alpha y,2\\alpha x)\\\\&=&\\alpha(-y,2x)\\\\&=&\\alpha T(v)\\end{array}<\/span>\n
Por consiguiente, T<\/span> es una transformaci\u00f3n lineal.\n<\/p>\n
\n
Enlaces de inter\u00e9s<\/strong><\/p>\n
Clase Online<\/a>\nSocrative Student<\/a>\nPlataforma SIDWeb<\/a>\nReferencias Bibliogr\u00e1ficas<\/a><\/pre>\n","protected":false},"excerpt":{"rendered":"
Definici\u00f3n. Sean y espacios vectoriales cualesquiera. Una transformaci\u00f3n lineal es una funci\u00f3n que asigna a cada vector un vector \u00fanico y que satisface las siguientes condiciones: . . Ejemplo. Sean y espacios vectoriales. Demuestre que la transformaci\u00f3n definida por es lineal. Soluci\u00f3n. . Por consiguiente, es una transformaci\u00f3n lineal. \u00a0 Teorema. Sea una transformaci\u00f3n lineal … Sigue leyendo cl3-01. Transformaciones Lineales<\/span><\/a><\/p>\n","protected":false},"author":609,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[1414634],"tags":[],"class_list":["post-2002","post","type-post","status-publish","format-standard","hentry","category-temas-2da-evaluacion"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/posts\/2002","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/users\/609"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/comments?post=2002"}],"version-history":[{"count":0,"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/posts\/2002\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/media?parent=2002"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/categories?post=2002"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/matg1049\/wp-json\/wp\/v2\/tags?post=2002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}