1.     Arrangements Of Numbers. 4

2.     The Ordered Pair 4

3.     The Ordered Nsome. 4

4.     The One Dimensional Matrix. 5

5.     The Inner Product Of Ordered Pairs (Twosomes) 5

6.     The Inner Product Of Matrices. 7

7.     Linear Combinations Of Basis Matrices. 9

8.     Reversing Linear Combinations Of Basis Matrices. 10

9.     From Matrices To Functions. 10

10.       Basis Functions With Negative Or Real Indices. 13

11.       Problems Of Functions Whose Domain is  \mathbb{R} ... 15

12.       Creating The Inner Product Of Two Functions In  \mathbb{R} ... 15

 


Figure 1 Function Example Using CCP. 12

Figure 2 1 Dimensional Basis Matrices Of Cardinality 5. 13

Figure 3 1 Dimensional Basis Matrices Of Countably Infinite Cardinality. 14

Figure 4 1 Dimensional Basis Matrices Of Uncountably Infinite Cardinality. 15

 


Table 1 Function Example 1. 10

Table 2 Basis Function ↔ [1,0] 11

Table 3 Basis Function ↔ [01] 11

Table 4 Resulting Function ↔ 3.2 * [1,0] + 6 * [0,1] 11


This presentation was inspired in part by the Laws of Form by G. Spencer-Brown.

1.   Arrangements Of Numbers

Numbers can be arranged in any way.

Example 1

6,7,8,256

Example 2:

1, 3, 5,

4, 5

6

Example 3:

356, 1000,

3, 456,

123 , 666

2.   The Ordered Pair

An ordered pair is an arrangement of two numbers in which one precedes the other, hence the term “ordered”. For example,

1,0

0,1

223,356

An ordered pair is also an ordered twosome.

3.   The Ordered Nsome

An ordered Nsome is an arrangement of N numbers in which each number precedes another save the last number, hence the term “ordered”. For example,

Example 1: An ordered foursome:

2,4,5,6

Example 2: An ordered sixsome

345,2,2.55,0.004,55,123

4.   The One Dimensional Matrix

An ordered Nsome is a matrix, when an index added. The index can be anything, but is usually limited to numbers, especially natural numbers, i.e. 0, 1, 2, 3, …

Ordered pair 1, 0 as a matrix

index

value

0

1

1

0

Ordered pair 0, 1 as a matrix

index

value

0

0

1

1

An ordered foursome as a matrix.

index

value

0

2

1

365

2

3.1415...

3

6.6

In this presentation we will concern ourselves exclusively to one dimensional objects.

5.   The Inner Product Of Ordered Pairs (Twosomes)

The inner product of two ordered Nsome results in a third ordered Nsome.

1.      The product of the first number of both ordered Nsomes is the first number of the third ordered Nsome.

2.      The product of the second number of both odered Nsomes is the second number of the third ordered Nsome.

3.      The product of the nth number of both odered Nsomes is the nth number of the third ordered Nsome.

Example 1

ordered twosome 1

3, 4

ordered twosome 2

2, 6

ordered twosome 3

3x2 = 6

4x6 = 24

6, 24

 

Example 2

ordered twosome 1

1, 0

ordered twosome 2

0, 1

ordered twosome 3

1x0 = 0

0x1 = 0

0,0

We denote the inner product with the operator . In the example above

Note: In example 2, when the result of the inner product = 0,0, the first and second ordered twosomes are orthogonal if the inner product of the ordered twosomes with themselves is not 0,0.

The inner product of the ordered twosome 1,0 with itself is

and the inner product of the ordered twosome 0,1 with itself is

Since these inner products are not equal to 0,0, 1,0 and 0,1 are orthogonal ordered twosomes.

6.   The Inner Product Of Matrices

The inner product of two ordered Nsomes can easily be applied to one dimensional matrices which are Nsomes with indices:

1.      The product of the first number of both one dimensional matrices is the first number of the third one dimensional matrix.

2.      The product of the second number of both one dimensional matrices is the second number of the third one dimensional matrix.

3.      The product of the nth number of both one dimensional matrices is the nth number of the third one dimensional matrix.

Example 1

one dimensional matrix 1

index

value

0

3

1

4

one dimensional matrix 2

index

value

0

2

1

6

one dimensional matrix 3

index

value

0

3 x 2 = 6

1

4 x 6 = 24

 

Example 2

one dimensional matrix 1

index

value

0

1

1

0

one dimensional matrix 2

index

value

0

0

1

1

one dimensional matrix 3

index

value

0

1x0 = 0

1

0x1 = 0

We denote the inner product with the operator  with ordered Nsomes. In the example above except using square brackets for matrices:

Note: In example 2, when the result of the inner product = 0,0, the first and second one dimensional matrices are orthogonal if the inner product of the one dimensional matrices with themselves is not 0,0.

The inner product of the one dimensional matrix 1,0 with itself is

and the inner product of the one dimensional matrix 0,1 with itself is

Since these inner products are not equal to 0,0, 1,0 and 0,1 are orthogonal one dimensional matrices.

These ideas can be extended to matrices of one dimension and high cardinality, i.e. the number of elements in the matrix:

etc.

7.   Linear Combinations Of Basis Matrices

A set of orthogonal matrices form a basis. A linear combination of basis matrices can generate other matrices of the same cardinality. A linear combination of basis matrices is defined as

c  =

where

c = the resulting matrix

an = coefficient n

bn = basis vector n

N = the cardinality of the matrix

Any one dimensional matrix of cardinality N can be created by a linear combination of basis vectors of cardinality N. A different way of stating this is that the basis spans (a term from matrix theory) the matrix space.

Example:

b0 = [1,0]

b1 = [0,1]

a0 = 3.2

a1 = 6

8.   Reversing Linear Combinations Of Basis Matrices

We can also determine an by taking the inner product of the corresponding basis function, bn with c:

 

 

From the example above:

b0 = [1,0]

c = [3.2,6]

 

We can also describe the coefficiants an as the degree to which c resembles bn or the degree to which bn is present in c.  Note: To be completely rigerous and to assist in understanding the extension from matrices to continuous functions in  \mathbb{R} , the real numbers, we should include two indices for the matrices:

 

bnm

an

cm

 

Where

n = the nth basis matrix

m = the mth element of the matrix

 

So the above should be written

 

 

Note also that the an are scaler values, not matrices, hence no m index.

9.   From Matrices To Functions

A function is a mapping from a domain to a range. Only one value in the range can be assigned a value in the domain. For example, the table shows the domain by the values of x and the range by the values of F(x). Each value of x is assigned a value F(x).

x

F(x)

0

2

1

365

2

3.1415

3

6.6

 

Table 1 Function Example 1

 

This is identical to the index and values of the matrix shown in section 4. If we equate the domain of the function with the index of the matrix, we can use the matrix properties of orthogonal basis and linear combinations to analyze functions using basis functions, the functional analog of basis matrices.  The following figure shows how to illustrate a linear combination of the basis matrices [1,0] and [0,1] to create the function, illustrated in the

Cartresian coordinate plane (CCP), shown in Table 4.  Note: There are two basis functions shown in Table 2 and Table 3, each of which is mapped on the x-coordinates 0

and 1 in the CCP.

 

 

x

F(x)

0

1

1

0

 

 

Table 2 Basis Function [01]

 

 

 

x

F(x)

0

0

1

1

 

Table 3 Basis Function ↔ [01]

 

 

x

F(x)

0

3.2

1

6

 

Table 4 Resulting Function ↔ 3.2 * [1,0] + 6 * [0,1]

Figure 1 Function Example Using CCP

 

 

Figure 2 1 Dimensional Basis Matrices Of Cardinality 5

 

As shown in Figure 2, this idea can be extended to more basis functions (matrices), even an infinite number, extending to all x = 0, 1, 2, 3, ....  All the the basis functions (matrices) still have the following properties:

  1. The inner product of 2 different basis functions (matrices) is 0
  2. The inner product of  the basis function (matrix) with itself is 1
  3. Any function whose domain is the natural numbers, \mathbb{N}, can be expressed as a linear combination of the basis functions (matrices).
  4. We can determine an by taking the inner product of the corresponding basis function, bn with c, the function.

10.         Basis Functions With Negative Or Real Indices

Just as we can extend the basis functions (matrices) domain (indices) to all the natural numbers, when can extend it to the integers as well:

Figure 3 1 Dimensional Basis Matrices Of Countably Infinite Cardinality

 

As shown in Figure 3, all the the basis functions (matrices) still have the following properties:

  1. The inner product of 2 different basis functions (matrices) is 0
  2. The inner product of  the basis function (matrix) with itself is 1
  3. Any function whose domain is the integers, \mathbb{Z}, can be expressed as a linear combination of the basis functions (matrices).
  4. We can determine an by taking the inner product of the corresponding basis function, bn with c, the function.

 

Just as we can extend the basis functions (matrices) domain (indices) to all the integers, when can extend it to the real numbers as well:

 

Figure 4 1 Dimensional Basis Matrices Of Uncountably Infinite Cardinality

 

As shown in Figure 4, all the the basis functions (matrices) still have the following properties:

  1. The inner product of 2 different basis functions (matrices) is 0
  2. The inner product of  the basis function (matrix) with itself is 1
  3. Any function whose domain is the real,  \mathbb{R} , can be expressed as a linear combination of the basis functions (matrices).
  4. We can determine an by taking the inner product of the corresponding basis function, bn with c, the function.

11.         Problems Of Functions Whose Domain is  \mathbb{R}

Obviously, the examples presented in the preceding sections become difficult when basis matix cardinality is very large, even infinite as in the case with basis functions whose domain is  \mathbb{R} .  However, integral calculus provides a means by which these functions can be manipulated quite easily.  In fact, extending the ideas of orthogonality, the inner product and linear combinations of bases from simple ordered pairs to continuous functions in  \mathbb{R}  is one of the greatest advancements in mathematics that has occurred in the past 150 years.

 

12.         Creating The Inner Product Of Two Functions In  \mathbb{R}

The following equations and xxx illustrate the very simple idea of the integral in calculus.  Simple ideas, however, can have complex mathematical principles attached. 

 

 

All the fancy math notation “simply” means the integral is a sum of rectangles that approximate the area under a funtion, the curve shown in figure , in this case the product of two functions F1(x) and F2(x), when Δ<0 and the number of elements in the sum = (M-L)/Δ.  The relationship between the number of “samples”, N, i.e. rectangles and the distance between M and L is

                                                                                                                          

Also,

As Δ→0, the number of samples increases exponentially until they reach an infinitely accurate approximation of the area under the curve = F1(x) F2(x).

 

The integral of the product of  two funtions also has another property: As Δ→0, it also reaches an infinitely accurate approximation of the inner product of the value of both functions for each real number!  Note the ...+ F1(xn) F2(xn) + F1(xn+1) F2(xn+1) +... terms above are exactly the same as the terms in the simple cases of matrix inner product presented in section 6 and converge to the inner product of each function for each real number.

 

Real Basis Functions

The basis function shown in Figure 4 is interesting as an illustration, but has no real practical pupose.  There is very little information contained in a funcion which is 1 for 1 real number and 0 for all the rest.  Are there any functions which met our set of requirements:

 

  1. The inner product of 2 different basis functions (matrices) is 0
  2. The inner product of  the basis function (matrix) with itself is 1
  3. Any function whose domain is the real,  \mathbb{R} , can be expressed as a linear combination of the basis functions (matrices).
  4. We can determine an by taking the inner product of the corresponding basis function, bn with c, the function.

 

Translating this to the language of calculus we obtain

 

 

 

The answer is yes:

 

 

Where Fb(x,y)↔e-2πjxξ

yξ

 

Note that referring to section 8, the y/z  and x variables correspond to the matrix indices n and m

13.         Sinusoidal Basis Functions In  \mathbb{R}

The rather formadable formulas and the end of the last section are actually structured simply.  The complex expoential is, of course, Eulers astonishing discovery of the relationship with the transcendental constant e = ln-1(1) =

 

2.71828182845904523536028747135266249775724709369995...

 

and the sine and cosine functions.

 

The famous formula is

 

 

where

 

ξ = radians

 

By scaling the argument with x or t, which are real variables, to create a function we have