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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

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
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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