Skip to content Skip to navigation

Connexions

You are here: Home » Content » Producto Interno

Navigation

Lenses

What is a lens?

Definition of a lens

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.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Señales y Sistemas"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Señales y Sistemas"

    Comments:

    "Señales y Sistemas is a Spanish translation of Dr. Rich Baraniuk's collection Signals and Systems (col10064). The translation was coordinated by an an assistant electrical engineering professor […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Producto Interno

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

Based on: Inner Products by Michael Haag, Justin Romberg

Summary: Este modulo describe el concepto de producto interno, el cual nos da una introducción a los espacios de HIlbert. Ejemplos y propiedades de estos dos conceptos son dicutidos.

Definición: Producto Interno

De seguro ya tiene una idea del producto interno, también conocido como producto punto, en Rn n de alguno de sus cursos de matemáticas o de cómputo. Si no, definiremos el producto interno de la siguiente manera, tenemos dados algunas x∈Rn x n y y∈Rn y n

Definition 1: Producto Interno
El producto interno esta definido matemáticamente de la siguiente manera:
〈x,y〉=yT⁢x=( y 0 y 1 … y n − 1 )⁢ x 0 x 1 ⋮ x n − 1 =∑ i =0n−1 x i ⁢ y i x y y x y 0 y 1 … y n − 1 x 0 x 1 ⋮ x n − 1 i n 1 0 x i y i (1)

Producto Interno en 2-D

Si tenemos x∈R2 x 2 y y∈R2 y 2 , entonces podemos escribir el producto interno como:

〈x,y〉=∥x∥⁢∥y∥⁢cosθ x y x y θ (2)
donde θθ es el ángulo entre xx y yy.

Figura 1: Gráfica general de vectores y ángulos mencionados en las ecuaciones anteriores.
Figura 1 (inprod_f1.png)

Geométricamente, el producto interno nos dice sobre la fuerza de xx en la dirección de yy.

Ejemplo 1

Por ejemplo, si ∥x∥=1 x 1 , entonces 〈x,y〉=∥y∥⁢cosθ x y y θ

Figura 2: Gráfica de los dos vectores del ejemplo anterior.
Figura 2 (inprod_f2.png)

Las siguientes características son dadas por el producto interno:

  • 〈x,y〉 x y mide la longitud de la proyección de yy sobre xx.
  • 〈x,y〉 x y es el máximo (dadas ∥x∥ x , ∥y∥ y ) donde xx y yy estan en la misma dirección ( (θ=0)⇒(cosθ=1) θ 0 θ 1 ).
  • 〈x,y〉 x y es cero cuando (cosθ=0)⇒(θ=90°) θ 0 θ 90° , es decir xx y yy son ortogonales.

Reglas del Producto Interno

En general el producto interno en un espacio vectorial complejo es solo una función (tomando dos vectores y regresando un número complejo) que satisface ciertas condiciones:

  • Simetria Conjugada: 〈x,y〉=〈x,y〉¯ x y x y
  • Escalado: 〈α⁢x,y〉=α⁢〈(x,y)〉 α x y α x y
  • Aditividad: 〈x+y,z〉=〈x,z〉+〈y,z〉 x y z x z y z
  • "Positividad": ∀ x ,x≠0:〈x,x〉>0 x x 0 x x 0
Definition 2: Ortogonal
Decimos que xx y yy son ortogonales si: 〈x,y〉=0 x y 0

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

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.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks