{"id":699,"date":"2018-03-13T19:24:03","date_gmt":"2018-03-13T22:24:03","guid":{"rendered":"http:\/\/www.professores.uff.br\/jcolombo\/?page_id=699"},"modified":"2018-07-05T11:21:59","modified_gmt":"2018-07-05T14:21:59","slug":"algebra-linear-ii-gan00164-turma-b1-12018","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/jcolombo\/algebra-linear-ii-gan00164-turma-b1-12018\/","title":{"rendered":"\u00c1lgebra Linear II- GAN00164 \u2013 Turma B1 \u2013 1\/2018"},"content":{"rendered":"<p><strong>Carga hor\u00e1ria:<\/strong> 68 hs, turma B1<\/p>\n<p><strong>Exerc\u00edcios &amp; fatos interessantes:\u00a0<\/strong><\/p>\n<ol>\n<li>Prove que isso realmente deve acontecer: as proje\u00e7\u00f5es ortogonais do cubo unit\u00e1rio sobre dois sub espa\u00e7os complementares sempre t\u00eam, numericamente, as mesmas medidas. Veja o <a href=\"https:\/\/www.geogebra.org\/m\/Hntm8Bjy\">arquivo no geogebra<\/a> que o Prof. Humberto Bortolossi criou para mostrar a propriedade &#8211; veja no <a href=\"https:\/\/www.facebook.com\/groups\/180800351982211\/?multi_permalinks=1747767638618800%2C1747592408636323&amp;notif_id=1521038565431759&amp;notif_t=group_activity&amp;ref=notif\">face<\/a> .<\/li>\n<\/ol>\n<p><strong>Informa\u00e7\u00f5es:<\/strong><br \/>\nAqui est\u00e3o alguns arquivos em pdf que baixei da rede e que tem aplica\u00e7\u00f5es de \u00e1lgebra linear. Caso algu\u00e9m queira ler algum destes arquivos e esteja encontrando dificuldades eu teria muito prazer ajudar, \u00e9 s\u00f3 me procurar.<\/p>\n<ul>\n<li>Mudan\u00e7a de coordenadas do sistema CIE-RBG para o sistema CIE-XYZ. (<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2017\/09\/1-2017-sistemas_de_coordenadas_de_cores.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">pdf<\/a>)<\/li>\n<li>Uma aplica\u00e7\u00e3o de \u00e1lgebra linear a <a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2017\/09\/1-2017-Tomografia.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">tomografia<\/a>. Este trabalho foi apresentado pela Danielle Nascimento monitora de \u00e1lgebra linear no XIV Semana de Monitoria.<\/li>\n<li>Utiliza\u00e7\u00e3o de \u00e1lgebra linear em Circuitos El\u00e9tricos. (<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2017\/09\/1-2017-curto_circuitos.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">pdf<\/a>)<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Monitoria:<\/strong><\/p>\n<hr \/>\n<p><strong>Lista de Exerc\u00edcios para a P1<\/strong><br \/>\nLista 1 &#8211; Exerc\u00edcios cap\u00edtulo 8 da refer\u00eancia [3], p\u00e1g. 295-299.<br \/>\nLista 2 &#8211; Exerc\u00edcios cap\u00edtulo 6 (Determinantes) da refer\u00eancia [0], p\u00e1g. 160-161.<br \/>\nLista 3 &#8211; Exerc\u00edcios cap\u00edtulo 9 (Diagonaliza\u00e7\u00e3o) da refer\u00eancia [3], p\u00e1g. 314-332.<br \/>\nLista 4 &#8211; Exerc\u00edcios cap\u00edtulo 10 (10.29 at\u00e9 10.77) da refer\u00eancia [3], p\u00e1g. 350-356.<br \/>\n<strong>Lista de Exerc\u00edcios para a P2<\/strong><br \/>\nLista 1 &#8211; Exerc\u00edcios cap\u00edtulo 9 (Produto Interno, Proje\u00e7\u00f5es e Operadores Ortogonais) da refer\u00eancia [0], p\u00e1g.212-216.<br \/>\nLista 2 &#8211; Cap\u00edtulo 11 da refer\u00eancia [6] exerc\u00edcios 11.1 at\u00e9 o 11.10 e do 11.18 at\u00e9 11.31<br \/>\nLista 3 &#8211; Cap\u00edtulo 14\u00a0da refer\u00eancia [6] exerc\u00edcios 14.1 at\u00e9 14.4 e 14.8 at\u00e9 14.23.<\/p>\n<hr \/>\n<p><strong>Hor\u00e1rio:<\/strong> 3\u00aa e 5\u00aa-feiras na sala IM-304G das 20 \u00e0s 22hs.<br \/>\n<strong>Meu hor\u00e1rio de atendimento:<\/strong> 4\u00aa-feiras, das 14-18hs em meu gabinete.<\/p>\n<hr \/>\n<p><strong>Provas<\/strong><br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/P1AlgLin_II.pdf\">P1<\/a> &#8211; 08\/05\/2018<br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/P2_AlgLin_II.pdf\">P2<\/a> &#8211; 03\/07\/2018<br \/>\nVR &#8211; 10\/07\/2018<br \/>\nVS &#8211; 12\/07\/2018<\/p>\n<hr \/>\n<p style=\"text-align: center\"><strong>Cronograma<\/strong><\/p>\n<p><strong>Aula \u00a0 \u00a0 Data\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Assunto<\/strong><br \/>\n01\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 13\/03\u00a0\u00a0\u00a0\u00a0 Apresenta\u00e7\u00e3o \u00a0da \u00a0disciplina, ementa, bibliografia e Avalia\u00e7\u00e3o. Determinantes exerc\u00edcios<br \/>\n02\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 15\/03\u00a0\u00a0\u00a0\u00a0 Determinantes: F\u00f3rmula e propriedades b\u00e1sicas<br \/>\n03\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 20\/03\u00a0\u00a0\u00a0 Determinantes: det(AB)= det(A)det(B), Adjunta cl\u00e1ssica, outra f\u00f3rmula para o determinante, determinante e \u00e1rea.<br \/>\n04\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a022\/03\u00a0\u00a0 \u00a0 Diagonaliza\u00e7\u00e3o &#8211; Sub espa\u00e7os fixos, extra\u00e7\u00e3o de ra\u00edzes de operadores, pot\u00eancia de um operador.<br \/>\nXX\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 27\/03\u00a0\u00a0\u00a0\u00a0 Feriado<br \/>\n05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a003\/04 \u00a0\u00a0\u00a0\u00a0 Polin\u00f4mio caracter\u00edstico, Teorema de Cayley-Hamilton, problemas com o m\u00e9todo de diagonaliza\u00e7\u00e3o.<br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/Aula-6-Forma-de-Jordan.pdf\">06\u00a0<\/a> \u00a0\u00a0\u00a0\u00a0\u00a0 05\/04\u00a0\u00a0\u00a0 Forma de Jordan para matrizes 2X2, Polin\u00f4mio M\u00ednimo.<br \/>\n07 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a010\/04\u00a0\u00a0\u00a0\u00a0 Operadores de Proje\u00e7\u00e3o, decomposi\u00e7\u00e3o prim\u00e1ria I.<br \/>\n08 \u00a0\u00a0\u00a0\u00a0\u00a0 12\/04\u00a0\u00a0 \u00a0 Decomposi\u00e7\u00e3o prim\u00e1ria II &#8211;\u00a0 operadores diagonaliz\u00e1veis.<br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/Aula_09_autovalores_complexos.pdf\">09<\/a>\u00a0 \u00a0 \u00a0 17\/04\u00a0\u00a0\u00a0 \u00a0 Estudo de operadores com auto valores complexos + Exerc\u00edcios.<br \/>\n10\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 19\/04\u00a0\u00a0\u00a0\u00a0 Forma can\u00f4nica de Jordan.<br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/Aula_11_DP_FJ.pdf\">11<\/a>\u00a0\u00a0 \u00a0 \u00a0 \u00a0 24\/04\u00a0\u00a0\u00a0\u00a0 Decomposi\u00e7\u00e3o prim\u00e1ria e forma can\u00f4nica de Jordan.<br \/>\n12 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a026\/04\u00a0\u00a0\u00a0\u00a0 Aula de exerc\u00edcios.<br \/>\nxx \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a001\/05\u00a0\u00a0\u00a0\u00a0 Feriado &#8211; Dia do trabalhador.<br \/>\n13\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a003\/05\u00a0\u00a0 \u00a0 Aula de exerc\u00edcios<br \/>\n14\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a008\/05\u00a0\u00a0 \u00a0 <a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/P1AlgLin_II.pdf\">P1<\/a><br \/>\n15\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 10\/05 \u00a0 \u00a0\u00a0Corre\u00e7\u00e3o da P1 &#8211; n\u00e3o teve aula.<br \/>\n16\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 15\/05\u00a0\u00a0\u00a0\u00a0 Produto interno &#8211;<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-787\" src=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/proj-282x300.jpg\" alt=\"\" width=\"282\" height=\"300\" srcset=\"https:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/proj-282x300.jpg 282w, https:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/proj-768x818.jpg 768w, https:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/proj.jpg 890w\" sizes=\"auto, (max-width: 282px) 100vw, 282px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>17 \u00a0 \u00a0 \u00a0\u00a0 17\/05 \u00a0 \u00a0 Proje\u00e7\u00f5es, simetrias no plano e Gram-Schmidt<br \/>\n<a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/Aula_18_Dual.pdf\">18<\/a> \u00a0\u00a0\u00a0\u00a0\u00a0 22\/05 \u00a0 \u00a0\u00a0Dualidade, transposta e Adjunta.<br \/>\nXX \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 29\/05\u00a0\u00a0\u00a0\u00a0\u00a0 N\u00e3o teve aula \u2013 greve dos caminhoneiros<br \/>\nXX \u00a0\u00a0 \u00a0 \u00a0 31\/05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 N\u00e3o teve aula \u2013 greve dos caminhoneiros<br \/>\n20 \u00a0 \u00a0 \u00a0 \u00a0 05\/06 \u00a0\u00a0\u00a0\u00a0 Adjunta\u00a0\u00a0 (Prof. S\u00e9rgio)<br \/>\n21\u00a0\u00a0\u00a0\u00a0\u00a0 07\/06\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 Operadores Auto Adjuntos (Prof. S\u00e9rgio)<br \/>\n22\u00a0\u00a0\u00a0\u00a0\u00a0 12\/06\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 Teorema Espectral (Prof. S\u00e9rgio)<br \/>\n23\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 14\/06\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 N\u00e3o teve aula<br \/>\n24\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 19\/06\u00a0\u00a0\u00a0\u00a0 \u00a0 + sobre operadores proje\u00e7\u00e3o e <a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2018\/03\/Operadores-Ortogonais-no-espa\u00e7o.pdf\">Operadores Ortogonais no espa\u00e7o<\/a>.<br \/>\n25\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 21\/06 \u00a0 \u00a0\u00a0\u00a0\u00a0Aula de Exerc\u00edcios + formas bi-lineares e classifica\u00e7\u00e3o de c\u00f4nicas<br \/>\n26\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 26\/06\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Aula de Exerc\u00edcios<br \/>\n27\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 28\/06\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Aula de Exerc\u00edcios<br \/>\n28\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 03\/07\u00a0 \u00a0 \u00a0\u00a0\u00a0 P2<br \/>\n29 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 05\/07\u00a0 \u00a0 \u00a0\u00a0\u00a0 Corre\u00e7\u00e3o da P2, vista da P2 + detalhes sobre a VR.<\/p>\n<hr \/>\n<p><strong>Bibliografia:<\/strong><br \/>\n[0] <a href=\"http:\/\/www.professores.uff.br\/jcolombo\/wp-content\/uploads\/sites\/124\/2017\/09\/1-2017-NotasColomboKoiller_EM_PREPARACAO.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">\u00c1lgebra Linear: Texto em prepara\u00e7\u00e3o<\/a>, Jones Colombo e Jos\u00e9 Koiller.<br \/>\n[1] \u00c1lgebra Linear, Boldrini, Costa, Ribeiro,Wetzler, Editora Harbra, 3\u00aa edi\u00e7\u00e3o.<br \/>\n[2] Contemporary Linear Algebra, Anton,H, Busby, Robert C., John Wiley &amp; Sons, Inc.<br \/>\n[3] \u00c1lgebra Linear, S. Lipschutz; M.L. Lipson, Cole\u00e7\u00e3o Schaum, Bookman, 4\u00aa edi\u00e7\u00e3o, 2011.<br \/>\n[4] \u00c1lgebra Linear e Aplica\u00e7\u00f5es, Carlos A. Callioli , Hygino H. Domingues, Roberto C.F. Costa, Editora Atual,6\u00aa edi\u00e7\u00e3o, 1990.<br \/>\n[5] \u00c1lgebra Linear, S. Lang, Editora Edgar Blucher Ltda, 1971.<br \/>\n[6] \u00c1lgebra Linear, E.L. Lima, Cole\u00e7\u00e3o Matematica Universit\u00e1ria, 2\u00aa edi\u00e7\u00e3o, 1996.<br \/>\n[7] \u00c1lgebra Linear, Hoffman, K., Kunze, R. Prentice-Hall, INC, Englewood Cliff, New Jersey, 1971.<\/p>\n<hr \/>\n<p><strong>Ementa:<br \/>\n<\/strong><br \/>\n1. Determinantes sobre corpos<br \/>\n1.1 Breve revis\u00e3o sobre o corpo dos n\u00fameros complexos, conjuga\u00e7\u00e3o e norma.<br \/>\n1.2 Determinantes de matrizes de ordem n sobre corpos (C, R ou Q). Propriedades. Express\u00f5es dos determinantes de matrizes de ordens 1, 2 ou 3. C\u00e1lculo do determinante por redu\u00e7\u00e3o por linhas.<br \/>\n1.3 Adjunta cl\u00e1ssica e cofatora. Propriedade. Regra de Cramer.<br \/>\n2. Autovalores e autovetores em espa\u00e7os vetoriais sobre R ou C<br \/>\n2.1 C-espa\u00e7o vetorial. Cn e Mnxn(C).<br \/>\n2.2 Subespa\u00e7os invariantes. Autovalores e autovetores: defini\u00e7\u00e3o e exemplos (n\u00e3o esquecer das fun\u00e7\u00f5es C\u221e: exponenciais, seno e coseno, importantes em EDO).<br \/>\n2.3 Determinante de operadores lineares em espa\u00e7os vetoriais sobre R ou C de dimens\u00e3o finita.Polin\u00f4mio caracter\u00edstico. Subespa\u00e7o caracter\u00edstico. Propriedades de autovalores e autovetores.<br \/>\n2.4 Operadores diagonaliz\u00e1veis. Multiplicidade alg\u00e9brica e multiplicidade geom\u00e9trica.<br \/>\n2.5 Teorema de Hamilton-Cayley. Aplica\u00e7\u00f5es.<br \/>\n3. Espa\u00e7os com produto interno.<br \/>\n3.1 Produto interno em espa\u00e7os vetoriais reais e complexos: defini\u00e7\u00e3o e exemplos.<br \/>\n3.2 Norma, desigualdade de Cauchy-Schwarz, desigualdade triangular. O conceito de \u00e2ngulo em espa\u00e7os vetoriais reais.<br \/>\n3.3 Conjuntos ortogonais, ortonormais. Bases ortogonais e ortonormais. Complemento ortogonal. Proje\u00e7\u00e3o ortogonal. Processo de ortonormaliza\u00e7\u00e3o de Gram-Schmidt.<br \/>\n4. Transforma\u00e7\u00f5es lineares em espa\u00e7os vetoriais de dimens\u00e3o finita com produto interno.<br \/>\n4.1 A adjunta de uma transforma\u00e7\u00e3o linear. Matriz da adjunta.<br \/>\n4.2 Operadores auto-adjuntos. Propriedades. Teorema espectral sobre R e C.<br \/>\n4.3 Transforma\u00e7\u00f5es lineares unit\u00e1rias e ortogonais. Isometrias. Classifica\u00e7\u00e3o das isometrias do plano.<br \/>\n4.4 Forma R-bi linear, forma quadr\u00e1tica. Aplica\u00e7\u00e3o: identifica\u00e7\u00e3o de c\u00f4nicas e quadr\u00e1ticas.<\/p>\n<hr \/>\n<p>\u00daltima revis\u00e3o em Junho, 2018.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Carga hor\u00e1ria: 68 hs, turma B1 Exerc\u00edcios &amp; fatos interessantes:\u00a0 Prove que isso realmente deve acontecer: as proje\u00e7\u00f5es ortogonais do cubo unit\u00e1rio sobre dois sub espa\u00e7os complementares sempre t\u00eam, numericamente, as mesmas medidas. Veja o arquivo no geogebra que o Prof. Humberto Bortolossi criou para mostrar a propriedade &#8211; veja no face . Informa\u00e7\u00f5es: Aqui [&hellip;]<\/p>\n","protected":false},"author":129,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[],"tags":[],"class_list":["post-699","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/699","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/users\/129"}],"replies":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/comments?post=699"}],"version-history":[{"count":33,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/699\/revisions"}],"predecessor-version":[{"id":822,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/699\/revisions\/822"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/media?parent=699"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/categories?post=699"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/tags?post=699"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}