Resumen Algebra Lineal
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(93.18) Álgebra Lineal – ITBA
Copyright: Ing. Daniel Palombo 2008
Desde ApuntesITBA nos hemos tomado el trabajo de escanear y recopilar este material, con el afán de brindarles a los futuros ingenieros del ITBA los mejores
recursos a la hora de estudiar.
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Temas Matrices Producto de matrices Ley asociativa Leyes distributivas Matrices inversibles Matriz traspuesta Matriz simétrica Matriz asimétrica Matriz de probabilidad Potencias de matrices (naturales) Matriz nilpotente Matriz idempotente Matrices elementales Elementales inversas Cálculo de la inversa de una matriz Matrices triangulares Descomposición LU Resolución de sistemas de ecuaciones con LU Descomposición PA = LU Determinantes Menor de un elemento Cofactor de un elemento Determinante de una matriz Teoremas Matriz cofactor Matriz adjunta Propiedades de la sumatoria Propiedades de Vandermonde Espacios Vectoriales Reglas de cerradura Operación interna Operación externa Suma Producto Subespacios Vectoriales Subespacios triviales Teoremas Espacio vectorial afín Sistema de generadores
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Independencia lineal Espacio vectorial finitamente generado Bases Dimensión Coordenadas Base canónica Cambio de base (coordenadas) Procedimiento para pasar de la base canónica a otra base Producto Escalar Definición Norma de un vector Desigualdad de Cauchy Sistema ortogonal Sistema ortonormal Base ortonormal (BO) Método de construcción de una BO (Gram-Schmidt) Matriz ortogonal Proyección ortogonal sobre un subespacio Complemento ortogonal Suma directa Otra forma de representar a un producto escalar Aproximación por el método de los cuadrados mínimos Producto Hermítico Propiedades Ejemplo Producto hermítico estándar Productos escalares particulares Espacio vectorial de funciones continuas Espacio vectorial de polinomios Transformación Lineal Propiedades Definición Transformación Cero Transformación Identidad Cálculo de Nu(t) e Im(t) Cálculo de T conociendo Nu(t) Cálculo de T conociendo Im(t) Cálculo de T conociendo Nu(t) e Im(t) Representación matricial de una Transformación Lineal Regla de transformación
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Diagrama esquemático Isomorfismos Transformación inyectiva Transformación sobreyectiva (Hepimorfismo) Transformación biyectiva (Isomorfismo) Espacios vectoriales isomorfos Composición de Transformaciones Lineales Inversa de una transformación lineal Transformaciones lineales especiales Rotaciones Simetrías Expansiones y compresiones Cortes Isometrías Espacios vectoriales isométricamente isomorfos Autovalores y Autovectores Polinomio característico Matrices semejantes y diagonalización Transformaciones Hermíticas Diagonalización ortogonal Formas cuadráticas Hipercuadráticas Clasificación de cónicas Funciones pares e impares Polinomios matriciales Teorema de Cayley-Hamilton Polinomio mínimo Matrices Nilpotentes Matrices nilpotentes de orden p Forma de Jordan
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