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Inner Products

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: This module describes the concept of inner products, which leads into our introduction of Hilbert spaces. Examples and properties of both of these concepts are discussed.

Note: You are viewing an old version of this document. The latest version is available here.

Definition: Inner Product

You may have run across inner products, also called dot products, on Rn n before in some of your math or science courses. If not, we define the inner product as follows, given we have some x∈Rn x n and y∈Rn y n

Definition 1: inner product
The inner product is defined mathematically as:
〈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)

Inner Product in 2-D

If we have x∈R2 x 2 and y∈R2 y 2 , then we can write the inner product as

〈x,y〉=∥x∥⁢∥y∥⁢cosθ x y x y θ (2)
where θθ is the angle between xx and yy.

Figure 1: General plot of vectors and angle referred to in above equations.
Figure 1 (inprod_f1.png)

Geometrically, the inner product tells us about the strength of xx in the direction of yy.

Example 1

For example, if ∥x∥=1 x 1 , then 〈x,y〉=∥y∥⁢cosθ x y y θ

Figure 2: Plot of two vectors from above example.
Figure 2 (inprod_f2.png)

The following characteristics are revealed by the inner product:

  • 〈x,y〉 x y measures the length of the projection of yy onto xx.
  • 〈x,y〉 x y is maximum (for given ∥x∥ x , ∥y∥ y ) when xx and yy are in the same direction ( (θ=0)⇒(cosθ=1) θ 0 θ 1 ).
  • 〈x,y〉 x y is zero when (cosθ=0)⇒(θ=90°) θ 0 θ 90° , i.e. xx and yy are orthogonal.

Inner Product Rules

In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complex number) that satisfies certain rules:

  • Conjugate Symmetry: 〈x,y〉=〈x,y〉¯ x y x y
  • Scaling: 〈α⁢x,y〉=α⁢〈(x,y)〉 α x y α x y
  • Additivity: 〈x+y,z〉=〈x,z〉+〈y,z〉 x y z x z y z
  • "Positivity": ∀ x ,x≠0:〈x,x〉>0 x x 0 x x 0
Definition 2: orthogonal
We say that xx and yy are orthogonal if: 〈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

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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