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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

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.

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: b 0 b 1 =( 1 1 )( 1 -1 ) b 0 b 1 1 1 1 -1 Normalizado con la norma ℓ 2 ℓ 2 : b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -1 ) b ~ 1 1 2 1 -1 Normalizado con la norma ℓ 1 ℓ 1 : b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -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 ⁢ 0 1 ℓ 2 0 1 : b 0 =( 1 0 ) b 0 1 0 b 1 =( 0 1 ) b 1 0 1 〈 b 0 , b 1 〉=∑ 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: ( 1 1 )( 1 -1 )= h 0 h 1 1 1 1 -1 h 0 h 1 〈 h 0 , h 1 〉=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 ={1  if  i=j0  if  i≠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 δ⁢i−j δ i j .

Ejemplo 4: Ejemplo de Base Ortonormal #1

b 0 b 2 =( 1 0 )( 0 1 ) b 0 b 2 1 0 0 1

Ejemplo 5: Ejemplo de Base Ortonormal #2

b 0 b 2 =( 1 1 )( 1 -1 ) b 0 b 2 1 1 1 -1

Ejemplo 6: Ejemplo de Base Ortonormal #3

b 0 b 2 =12⁢( 1 1 )12⁢( 1 -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 ={1  if  i=k0  if  i≠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: b 0 b 1 =12⁢( 1 1 )12⁢( 1 -1 ) b 0 b 1 1 2 1 1 1 2 1 -1 representa x=( 3 2 ) x 3 2

Ejemplo 8: Serie de Fourier Levemente Modificada

Dada la base 1T⁢ei⁢ ω 0 ⁢n⁢t| n =−∞∞ n 1 T ω 0 n t en L 2 ⁢ 0 T L 2 0 T donde T=2⁢π ω 0 T 2 ω 0 . f⁢t=∑ n =−∞∞〈(f,ei⁢ ω 0 ⁢n⁢t)〉⁢ei⁢ ω 0 ⁢n⁢t⁢1T 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 ⁢n⁢t〉=1T⁢∫0Tf⁢t⁢ei⁢ ω 0 ⁢n⁢t¯d t =1T⁢∫0Tf⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t 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

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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