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Este modulo discute los diferentes tipos de base, lo que nos conduce a la definición de base ortonormal. Son dados algunos ejemplos de la base ortonormal y sus usos también son discutidos.

Base normada

Base Normada
una base b i donde cada b i tiene una norma unitaria
i i b i 1
El concepto de bases se aplica a todos los espacios vectoriales . El concepto de base normada se aplica solo a espacios normados .
También usted puede normalizar una base: solo multiplique cada vector de la base por una constante, tal que 1 b i

Dada la siguiente base: b 0 b 1 1 1 1 -1 Normalizado con la norma 2 : b ~ 0 1 2 1 1 b ~ 1 1 2 1 -1 Normalizado con la norma 1 : b ~ 0 1 2 1 1 b ~ 1 1 2 1 -1

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Base ortogonal

Base Ortogonal
una base b i en donde los elementos son mutuamente ortogonales i i j b i b j 0
El concepto de base ortogonal se aplica solo a los Espacios de Hilbert .

Base canónica para 2 , también referida como 2 0 1 : b 0 1 0 b 1 0 1 b 0 b 1 i 1 0 b 0 i b 1 i 1 0 0 1 0

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Ahora tenemos la siguiente base y relación: 1 1 1 -1 h 0 h 1 h 0 h 1 1 1 1 -1 0

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Base ortonormal

Colocando las dos secciones (definiciones) anteriores juntas, llegamos al tipo de base más importante yútil:

Base Ortonormal
Una base que es normalizada y ortogonal i i b i 1 i i j b i b j
podemos acortar los dos argumentos en uno solo: b i b j δ i j donde δ i j 1 i j 0 i j Donde δ i j re refiere a la función delta Kronecker que también es escrita como δ i j .

Ejemplo de base ortonormal #1

b 0 b 2 1 0 0 1

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Ejemplo de base ortonormal #2

b 0 b 2 1 1 1 -1

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Ejemplo de base ortonormal #3

b 0 b 2 1 2 1 1 1 2 1 -1

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La belleza de las bases ortonormales

Trabajar con las bases Ortonormales es sencillo.Si b i es una base ortonormal, podemos escribir para cualquier x

x i α i b i
Es fácil encontrar los α i :
x b i k α k b k b i k α k b k b i
En donde en la ecuación anterior podemos usar el conocimiento de la función delta para reducir la ecuación: b k b i δ i k 1 i k 0 i k
x b i α i
Por lo tanto podemos concluir con la siguiente ecuación importante para x :
x i x b i b i
Los α i 's son fáciles de calcular (sin interacción entres los b i 's)

Dada la siguiente base: b 0 b 1 1 2 1 1 1 2 1 -1 representa x 3 2

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Serie de fourier levemente modificada

Dada la base n 1 T ω 0 n t en L 2 0 T donde T 2 ω 0 . f t n f ω 0 n t ω 0 n t 1 T Donde podemos calcular el producto interior de arriba en L 2 como f ω 0 n t 1 T t T 0 f t ω 0 n t 1 T t T 0 f t ω 0 n t

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Sea b i una base ortonormal para un espacio de Hilbert H . Entonces, para cualquier x H podemos escribir

x i α i b i
donde α i x b i .
  • “Análisis”: descomponer x en términos de b i
    α i x b i
  • "Síntesis": construir x de una combinación de las b i
    x i α i b i

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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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