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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

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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