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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.

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 xRn x n and yRn y n

Definition 1: standard inner product
The standard inner product is defined mathematically as:
x,y=yTx=( y 0 y 1 y n 1 ) x 0 x 1 x n 1 = i =0n1 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 xR2 x 2 and yR2 y 2 , then we can write the inner product as

x,y=xycosθ 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=ycosθ 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: αx,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  ,   x0    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

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