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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

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

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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

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