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Tipos de Bases

Module by: Michael Haag, Justin Romberg. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Types of Basis by Michael Haag, Justin Romberg

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Summary: 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.

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Base Normada

Definition 1: Base Normada
una base b i b i donde cada b i b i tiene una norma unitaria
∀i,i∈ℤ:∥ b i ∥=1 i i b i 1 (1)

nota:

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 ∥ 1 b i

Ejemplo 1

Dada la siguiente base: b0b1=111-1 b 0 b 1 1 1 1 -1 Normalizado con la norma ℓ 2 ℓ 2 : b ~ 0=1211 b ~ 0 1 2 1 1 b ~ 1=121-1 b ~ 1 1 2 1 -1 Normalizado con la norma ℓ 1 ℓ 1 : b ~ 0=1211 b ~ 0 1 2 1 1 b ~ 1=121-1 b ~ 1 1 2 1 -1

Base Ortogonal

Definition 2: Base Ortogonal
una base b i b i en donde los elementos son mutuamente ortogonales ∀i,i≠j:< b i , b j >=0 i i j b i b j 0

nota:

El concepto de base ortogonal se aplica solo a los Espacios de Hilbert .

Ejemplo 2

Base canónica para ℝ 2 ℝ 2 , también referida como ℓ 2 01 ℓ 2 0 1 : b0=10 b 0 1 0 b1=01 b 1 0 1 <b0,b1>=∑i=01 b 0 i b 1 i=1×0+0×1=0 b 0 b 1 i 1 0 b 0 i b 1 i 1 0 0 1 0

Ejemplo 3

Ahora tenemos la siguiente base y relación: 111-1=h0h1 1 1 1 -1 h 0 h 1 <h0,h1>=1×1+1×-1=0 h 0 h 1 1 1 1 -1 0

Base Ortonormal

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

Definition 3: Base Ortonormal
Una base que es normalizada y ortogonal ∀i,i∈ℤ:∥ b i ∥=1 i i b i 1 ∀i,i≠j:< b i , b j > i i j b i b j

notación:

podemos acortar los dos argumentos en uno solo: < b i , b j >= δ i j b i b j δ i j donde δ i j =1ifi=j0ifi≠j δ i j 1 i j 0 i j Donde δ i j δ i j re refiere a la función delta Kronecker que también es escrita como δi−j δ i j .

Ejemplo 4: Ejemplo de Base Ortonormal #1

b0b2=1001 b 0 b 2 1 0 0 1

Ejemplo 5: Ejemplo de Base Ortonormal #2

b0b2=111-1 b 0 b 2 1 1 1 -1

Ejemplo 6: Ejemplo de Base Ortonormal #3

b0b2=1211121-1 b 0 b 2 1 2 1 1 1 2 1 -1

La belleza de las Bases Ortonormales

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

x=∑i α i b i x i α i b i (2)
Es fácil encontrar los α i α i :
<x, b i >=<∑k α k b k , b i >=∑k α k < b k , b i > x b i k α k b k b i k α k b k b i (3)
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 =1ifi=k0ifi≠k b k b i δ i k 1 i k 0 i k
<x, b i >= α i x b i α i (4)
Por lo tanto podemos concluir con la siguiente ecuación importante para xx:
x=∑i<x, b i > b i x i x b i b i (5)
Los α i α i 's son fáciles de calcular (sin interacción entres los b i b i 's)

Ejemplo 7

Dada la siguiente base: b0b1=1211121-1 b 0 b 1 1 2 1 1 1 2 1 -1 representa x=32 x 3 2

Ejemplo 8: Serie de Fourier Levemente Modificada

Dada la base 1Tⅇⅈ ω 0 nt |n=-∞∞ n 1 T ω 0 n t en L 2 0T L 2 0 T donde T=2π ω 0 T 2 ω 0 . ft=∑n=-∞∞<f,ⅇⅈ ω 0 nt>ⅇⅈ ω 0 nt1T 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 L 2 como <f,ⅇⅈ ω 0 nt>=1T∫0Tftⅇⅈ ω 0 nt¯dt=1T∫0Tftⅇ-ⅈ ω 0 ntdt 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

7.7.3.2 Expansión de una Base Ortonormal en un Espacio Hilbert

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

x=∑i α i b i x i α i b i (6)
donde α i =<x, b i > α i x b i .
  • “Análisis”: descomponer x x en términos de b i b i
    α i =<x, b i > α i x b i (7)
  • "Síntesis": construir x x de una combinación de las b i b i
    x=∑i α i b i x i α i b i (8)

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Lenses

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