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Types of Bases

By Michael Haag, Justin Romberg

Normalized Basis

Definition 1: Normalized Basis
a basis b i b i where each b i b i has unit norm
i ,iZ: b i =1 i i b i 1
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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 2 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 1 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 ,ij: 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 2 2 , also referred to as 2 0 1 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 ,iZ: b i =1 i i b i 1 i ,ij: 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  ij 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 δij 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
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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
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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  ik b k b i δ i k 1 i k 0 i k
x, b i = α i x b i α i
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Therefore, we can conclude the following important equation for xx:
x=i(x, b i ) b i x i x b i b i
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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 1Tei ω 0 nt| n = n 1 T ω 0 n t on L 2 0 T L 2 0 T where T=2π ω 0 T 2 ω 0 . ft= n =(f,ei ω 0 nt)ei ω 0 nt1T 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 nt=1T0Tftei ω 0 nt¯d t =1T0Tfte(i ω 0 nt)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 xH x H we can write

x=i α i b i x i α i b i
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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
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  • "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
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