resumen algebra uned

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CAPÍTULO 1: ESPACIOS VECTORIALES Definición Un cuerpo (no alabeado) cuyos elementos (escalares) se representan por letras griegas. Un grupo abeliano de elementos (vectores) notados con letras latinas. Se define un espacio vectorial  sobre el cuerpo  si se define una aplicación de en  tal que a todo par , haga corresponder un elemento  cumpliendo las siguientes propiedades - Distributividad con respecto a la adición de :  - Distributividad respecto a la adición de :  - Asociatividad de los elementos :  -  , siendo  el elemento neutro de . - Siendo  la operación de  - Siendo  y  las operaciones de  Ejemplo 1 Dado el conjunto *  Se definen las leyes siguientes: - 1. , tal que  - 2. , tal que si    Comprobar que  es un espacio vectorial sobre  1. Es inmediato comprobar que: o  es una ley de composición interna, que cumple las propiedades asociativa y conmutativa. o  es el elemento neutro o  es el opuesto o simétrico de  o Por tanto  tiene estructura de grupo abeliano. 2. Si  y  són números reales se cumplen las siguientes propiedades: a)  [ ]    Por tanto [ ]  Con esto se verifica la a sociatividad respecto al producto de escalares b)  [ ]    Por tanto [ ]  Con esto se verifica la distributividad respecto a la suma de factores c)     Por tanto  Con esto se comprueba la di stributividad con respecto a la suma de escalares d) Es evidente que:  Por tanto,  cumple las cuatro propiedades de la ley externa y como  era grupo abeliano  tiene estructura de espacio vectorial.

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