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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

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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.

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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>=yTx= 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: <αx,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

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Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions 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.

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