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GA e Algebra Linear - UM CURSO DE GEOMETRIA ANALTICA E...

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UM CURSO DE GEOMETRIA ANAL ´ ITICA E ´ ALGEBRA LINEAR Reginaldo J. Santos Departamento de Matem´atica-ICEx Universidade Federal de Minas Gerais http://www.mat.ufmg.br/~regi Julho 2007
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Um Curso de Geometria Anal´ ıtica e ´ Algebra Linear Copyright c 2007 by Reginaldo de Jesus Santos (070801) ´ E proibida a reproduc ¸ ˜ao desta publicac ¸ ˜ao, ou parte dela, por qualquer meio, sem a pr´evia autorizac ¸ ˜ao, por escrito, do autor. Editor, Coordenador de Revis˜ao, Supervisor de Produc ¸ ˜ao, Capa e Ilustrac ¸ ˜oes: Reginaldo J. Santos ISBN 85-7470-006-1 Ficha Catalogr´afica Santos, Reginaldo J. S237u Um Curso de Geometria Anal´ ıtica e ´ Algebra Linear / Reginaldo J. Santos - Belo Horizonte: Imprensa Universit´aria da UFMG, 2007. 1. ´ Algebra Linear 2. Geometria Anal´ ıtica I. T´ ıtulo CDD: 512.5 516.3
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Conte´udo Pref´acio vii 1 Matrizes e Sistemas Lineares 1 1.1 Matrizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Operac ¸ ˜oes com Matrizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Propriedades da ´ Algebra Matricial . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Aplicac ¸ ˜ao: Cadeias de Markov . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Apˆendice I: Notac ¸ ˜ao de Somat´orio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.2 Sistemas de Equac ¸ ˜oes Lineares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.1 M´etodo de Gauss-Jordan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.2.2 Matrizes Equivalentes por Linhas . . . . . . . . . . . . . . . . . . . . . . . . 50 1.2.3 Sistemas Lineares Homogˆeneos . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.2.4 Matrizes Elementares (opcional) . . . . . . . . . . . . . . . . . . . . . . . . . 57 iii
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iv Conte´udo 2 Invers˜ao de Matrizes e Determinantes 77 2.1 Matriz Inversa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.1.1 Propriedades da Inversa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.2 Matrizes Elementares e Invers˜ao (opcional) . . . . . . . . . . . . . . . . . . . 82 2.1.3 M´etodo para Invers˜ao de Matrizes . . . . . . . . . . . . . . . . . . . . . . . . 86 2.1.4 Aplicac ¸ ˜ao: Interpolac ¸ ˜ao Polinomial . . . . . . . . . . . . . . . . . . . . . . . . 96 2.1.5 Aplicac ¸ ˜ao: Criptografia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.2 Determinantes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.2.1 Propriedades do Determinante . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.2.2 Matrizes Elementares e o Determinante (opcional) . . . . . . . . . . . . . . . 128 Apˆendice II: Demonstrac ¸ ˜ao do Teorema 2.11 . . . . . . . . . . . . . . . . . . . . . . 136 3 Vetores no Plano e no Espac ¸o 142 3.1 Soma de Vetores e Multiplicac ¸ ˜ao por Escalar . . . . . . . . . . . . . . . . . . . . . . 144 3.2 Produtos de Vetores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.2.1 Norma e Produto Escalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.2.2 Projec ¸ ˜ao Ortogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.2.3 Produto Vetorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.2.4 Produto Misto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Apˆendice III: Demonstrac ¸ ˜ao do item (e) do Teorema 3.5 . . . . . . . . . . . . . . . . 226 4 Retas e Planos 229 4.1 Equac ¸ ˜oes de Retas e Planos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.1.1 Equac ¸ ˜oes do Plano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.1.2 Equac ¸ ˜oes da Reta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.2 ˆ Angulos e Distˆancias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Um Curso de Geometria Anal´ ıtica e ´ Algebra Linear Julho 2007
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Conte´udo v 4.2.1 ˆ Angulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 4.2.2 Distˆancias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5 Espac ¸os R n 300 5.1 Independˆencia Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.1.1 Os Espac ¸os R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.1.2 Combinac ¸ ˜ao Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 5.1.3 Independˆencia Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 5.1.4 Posic ¸ ˜oes Relativas de Retas e Planos . . . . . . . . . . . . . . . . . . . . . . 322 5.2 Subespac ¸os, Base e Dimens˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Apˆendice IV: Outros Resultados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 5.3 Produto Escalar em R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.3.1 Produto Interno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.3.2 Bases Ortogonais e Ortonormais . . . . . . . . . . . . . . . . . . . . . . . . 372 5.4 Mudanc ¸a de Coordenadas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 5.4.1 Rotac ¸ ˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 5.4.2 Translac ¸ ˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 5.4.3 Aplicac ¸ ˜ao: Computac ¸ ˜ao Gr´afica - Projec ¸ ˜ao Ortogr´afica . . . . . . . . . . . . . 393 6 Diagonalizac ¸ ˜ao 405 6.1 Diagonalizac ¸ ˜ao de Matrizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 6.1.1 Motivac ¸ ˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 6.1.2 Autovalores e Autovetores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 6.1.3 Diagonalizac ¸ ˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 6.2 Diagonalizac ¸ ˜ao de Matrizes Sim´etricas . . . . . . . . . . . . . . . . . . . . . . . . . 440 6.2.1 Motivac ¸ ˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Julho 2007 Reginaldo J. Santos
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vi Conte´udo 6.2.2 Matrizes Ortogonais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Apˆendice V: Autovalores Complexos . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 6.3 Aplicac
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