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Instituto Superior T´cnico e Departamento de Matem´tica a ´ Sec¸ao de Algebra e An´lise c˜ a

´ Apontamentos das Aulas Te´ricas de Algebra Linear o
1o Semestre 2007/2008

LEAmb, LEMat, LQ, MEBiol, MEQ
Paulo Pinto http://www.math.ist.utl.pt/˜ ppinto/

Conte´ do u
1 Sistemas de Equa¸oes Lineares e c˜ 1.1 Matrizes . . . . . . . . . . . . . 1.2 Sistemas de Equa¸oes Lineares . c˜ 1.3 Matrizes Elementares . . . . . . 1.4 A matriz inversa . . . . . . . . 2 Determinante 3 Espa¸os Lineares (Vectoriais) c 3.1 Subespa¸os lineares – exemplos: n´ cleo, espa¸o colunas e linhas de c u c 3.2 Independencia linear . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bases e dimens˜o de Espa¸os Lineares . . . . . . . . . . . . . . . a c 3.4 Coordenadas de um vector numa base . . . . . . . . . . . . . . . . C´lculo a . . . . . . . . . . . . . . . . . . . . Matricial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 6 13 15 19 23 uma matriz 26 . . . . . . 31 . . . . . . 33 . . . . . . 39 40 47 . . . . . . . . . . . . . . . . 59 61 62 66 70 71 71 71 71

4 Valores Pr´prios, Vectores Pr´prios e diagonaliza¸˜o de Matrizes o o ca 5 Produtos Internos 6 Transforma¸oes Lineares c˜ 6.1 Matriz mudan¸a de base . . . . . . . . . . . . . . . . . . . . . . . . . c c˜ c˜ 6.2 Representa¸ao matricial de uma transforma¸ao linear . . . . . . . . . 6.3 Transforma¸oes injectivas, sobrejectiva e bijectivas – equa¸oes lineares c˜ c˜ 6.4 Valores e vectores pr´prios de transforma¸oes lineares . . . . . . . . . o c˜

7 Algumas Aplica¸oes c˜ 7.1 Formas quadr´ticas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a ınimos quadrados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 M´ 7.3 Equa¸oes diferenciais ordin´rias . . . . . . . . . . . . . . . . . . . . . . . . . c˜ a 1

1
1.1

Sistemas de Equa¸oes Lineares e C´lculo Matricial c˜ a
Matrizes...

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