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Types of Basis

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

Summary: This module discusses the different types of basis that leads up to the definition of an orthonormal basis. Examples are given and the useful of the orthonormal basis is discussed.

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

Normalized Basis

Definition 1: Normalized Basis
a basis b i b i where each b i b i has unit norm
∀ i ,i∈Z:∥ b i ∥=1 i i b i 1 (1)

Note:

The concept of basis applies to all vector spaces. The concept of normalized basis applies only to normed spaces.
You can always normalize a basis: just multiply each basis vector by a constant, such as 1∥ b i ∥ 1 b i

Example 1

We are given the following basis: b 0 b 1 =( 1 1 )( 1 -1 ) b 0 b 1 1 1 1 -1 Normalized with ℓ 2 ℓ 2 norm: b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -1 ) b ~ 1 1 2 1 -1 Normalized with ℓ 1 ℓ 1 norm: b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -1 ) b ~ 1 1 2 1 -1

Orthogonal Basis

Definition 2: Orthogonal Basis
a basis b i b i in which the elements are mutually orthogonal ∀ i ,i≠j:〈 b i , b j 〉=0 i i j b i b j 0

Note:

The concept of orthogonal basis applies only to Hilbert Spaces.

Example 2

Standard basis for ℝ 2 ℝ 2 , also referred to as ℓ 2 ⁢ 0 1 ℓ 2 0 1 : b 0 =( 1 0 ) b 0 1 0 b 1 =( 0 1 ) b 1 0 1 〈 b 0 , b 1 〉=∑ i =01 b 0 ⁢i⁢ b 1 ⁢i=1×0+0×1=0 b 0 b 1 i 1 0 b 0 i b 1 i 1 0 0 1 0

Example 3

Now we have the following basis and relationship: ( 1 1 )( 1 -1 )= h 0 h 1 1 1 1 -1 h 0 h 1 〈 h 0 , h 1 〉=1×1+1×-1=0 h 0 h 1 1 1 1 -1 0

Orthonormal Basis

Pulling the previous two sections (definitions) together, we arrive at the most important and useful basis type:

Definition 3: Orthonormal Basis
a basis that is both normalized and orthogonal ∀ i ,i∈Z:∥ b i ∥=1 i i b i 1 ∀ i ,i≠j:〈 b i , b j 〉 i i j b i b j

Notation:

We can shorten these two statements into one: 〈 b i , b j 〉= δ i j b i b j δ i j where δ i j ={1  if  i=j0  if  i≠j δ i j 1 i j 0 i j Where δ i j δ i j is referred to as the Kronecker delta function and is also often written as δ⁢i−j δ i j .

Example 4: Orthonormal Basis Example #1

b 0 b 2 =( 1 0 )( 0 1 ) b 0 b 2 1 0 0 1

Example 5: Orthonormal Basis Example #2

b 0 b 2 =( 1 1 )( 1 -1 ) b 0 b 2 1 1 1 -1

Example 6: Orthonormal Basis Example #3

b 0 b 2 =12⁢( 1 1 )12⁢( 1 -1 ) b 0 b 2 1 2 1 1 1 2 1 -1

Beauty of Orthonormal Bases

Orthonormal bases are very easy to deal with! If b i b i is an orthonormal basis, we can write for any xx

x=∑i α i ⁢ b i x i α i b i (2)
It is easy to find the α i α i :
〈x, b i 〉=〈∑k α k ⁢ b k , b i 〉=∑k α k ⁢〈( b k , b i )〉 x b i k α k b k b i k α k b k b i (3)
where in the above equation we can use our knowledge of the delta function to reduce this equation: 〈 b k , b i 〉= δ i k ={1  if  i=k0  if  i≠k b k b i δ i k 1 i k 0 i k
〈x, b i 〉= α i x b i α i (4)
Therefore, we can conclude the following important equation for xx:
x=∑i〈(x, b i )〉⁢ b i x i x b i b i (5)
The α i α i 's are easy to compute (no interaction between the b i b i 's)

Example 7

Given the following basis: b 0 b 1 =12⁢( 1 1 )12⁢( 1 -1 ) b 0 b 1 1 2 1 1 1 2 1 -1 represent x=( 3 2 ) x 3 2

Example 8: Slightly Modified Fourier Series

We are given the basis 1T⁢ei⁢ ω 0 ⁢n⁢t| n =−∞∞ n 1 T ω 0 n t on L 2 ⁢ 0 T L 2 0 T where T=2⁢π ω 0 T 2 ω 0 . f⁢t=∑ n =−∞∞〈(f,ei⁢ ω 0 ⁢n⁢t)〉⁢ei⁢ ω 0 ⁢n⁢t⁢1T f t n f ω 0 n t ω 0 n t 1 T Where we can calculate the above inner product in L 2 L 2 as 〈f,ei⁢ ω 0 ⁢n⁢t〉=1T⁢∫0Tf⁢t⁢ei⁢ ω 0 ⁢n⁢t¯d t =1T⁢∫0Tf⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t f ω 0 n t 1 T t T 0 f t ω 0 n t 1 T t T 0 f t ω 0 n t

Orthonormal Basis Expansions in a Hilbert Space

Let b i b i be an orthonormal basis for a Hilbert space HH. Then, for any x∈H x H we can write

x=∑i α i ⁢ b i x i α i b i (6)
where α i =〈x, b i 〉 α i x b i .
  • "Analysis": decomposing x x in term of the b i b i
    α i =〈x, b i 〉 α i x b i (7)
  • "Synthesis": building x x up out of a weighted combination of the b i b i
    x=∑i α i ⁢ b i x i α i b i (8)

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