Symmetric bilinear forms

In summary: And to each element in ##F##, we just associate itself. So if we have a function ##f:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}##, it means that for each two elements in ##F##, we have an associated element in ##\mathbb{R}^2##.If ##f## is ##-##, then to each ordered pair ##(a,b)## with ##a,b\in F##, we associate ##-(a,b)## in ##F##. We write this as ##a-b##.So to each two elements in ##F##, we just associate a new element
  • #1
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When I start to read the the article called "symmetric bi-linear forms", I face the following sentence. But I don't understand what does the following sentence suggest. Could someone please help me here?

We will now assume that the characteristic of our field is not 2 (so 1 + 1 is not = to 0)
 
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  • #2
A field is a very general object. It is a set ##F## together with two functions ##+:F\times F\rightarrow F## and ##\cdot:F\times F\rightarrow F## satisfying:

  • ##a+(b+c) = (a+b)+c##
  • ##a+b=b+a##
  • There exists a ##0\in F## such that ##a+0 = 0+a = a##
  • For each ##a\in F##, there is an element ##b\in F## such that ##a+b=b+a=0##
  • ##a\cdot(b\cdot c) = (a\cdot b)\cdot c##
  • ##a\cdot b = b\cdot a##
  • There exists an ##1\in F## such that ##1\neq 0## and ##a\cdot 1 = 1\cdot a = a##
  • If ##a\neq 0##, then there exists a ##b\in F## such that ##a\cdot b =b\cdot a = 1##

The usual suspects, such as ##\mathbb{Q}##, ##\mathbb{R}## and ##\mathbb{C}## are fields. All of these fields satisfy that ##1+1+1+1+1+...+1## (n times) is nonzero. But this is not a general property of a field. Fields that have the property are said to be of characteristic 0.

For example, consider ##F=\{0,1\}## and define

[tex]1+0 = 0+1 = 1~\text{and}~1+1=0+0=0[/tex]

and

[tex]1\cdot 0 = 0\cdot 0 = 0\cdot 1 = 0~\text{and}~1\cdot 1 = 1[/tex]

This satisfies all the field axioms, but it has ##1+1=0##. We say that this field has characteristic 2.

In general, if a field satisfies ##1+1+1+...+1=0## (n times). Then the field is said to have characteristic ##n##. We can always show that ##n## is a prime number.

The main advantage of assuming that the field is not of characteristic ##0##, is that we can divide by ##1+1##. If we use the nice notation ##2=1+1##, then ##1/2## exists.
 
  • #3
micromass said:
A field is a very general object. It is a set ##F## together with two functions ##+:F\times F\rightarrow F## and ##\cdot:F\times F\rightarrow F##
Could you please explain it a bit more?Specially the following.

micromass said:
##+:F\times F\rightarrow F## and ##\cdot:F\times F\rightarrow F##
What do you mean by "+" and "."?
 
  • #4
The cartesian product ##F\times F## just means the set of all ordered pairs. Thus

[tex]F\times F = \{(a,b)~\vert~a,b\in F\}[/tex]

For example ##(0,0)\in \mathbb{R}\times\mathbb{R}## and ##(-1,2)\in \mathbb{R}\times \mathbb{R}##.

That ##f## is a function from ##F\times F## to ##F## just means that we associate with each element in ##F\times F##, an element in ##F##.

So given ##(a,b)## with ##a,b\in F##, we associate an element ##f(a,b)\in F##. Specifically, if ##f## is ##+##, then to each ordered pair ##(a,b)## with ##a,b\in F##, we associate ##+(a,b)## in ##F##. We write this as ##a+b##.

So to each two elements in ##F##, we just associate a new element in ##F##.
 
  • #5


The sentence is suggesting that the concept of symmetric bilinear forms will only be discussed in the context of fields that do not have a characteristic of 2. This means that in these fields, the number 1 added to itself will always equal 2, and will not equal 0. This assumption is important because it allows for certain mathematical properties and operations to hold true for symmetric bilinear forms in these fields.
 

1. What is a symmetric bilinear form?

A symmetric bilinear form is a mathematical function that takes two vectors as inputs and returns a scalar value. It is linear in both of its inputs and satisfies the property of symmetry, meaning that switching the order of the inputs does not change the resulting scalar value. This type of form is commonly used in linear algebra and has applications in geometry and physics.

2. How is a symmetric bilinear form represented?

A symmetric bilinear form can be represented by a matrix with the same number of rows and columns as the vector space it operates on. The elements of the matrix correspond to the scalar values obtained by applying the form to pairs of basis vectors. This matrix is symmetric, meaning that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column.

3. What is the difference between a symmetric bilinear form and a non-symmetric bilinear form?

The main difference between a symmetric bilinear form and a non-symmetric bilinear form is the property of symmetry. While a symmetric form satisfies this property, a non-symmetric form does not. This means that the order of inputs matters for a non-symmetric form, while it does not for a symmetric form. Non-symmetric forms are also known as skew-symmetric forms.

4. What are some examples of symmetric bilinear forms?

One example of a symmetric bilinear form is the dot product in Euclidean space. Another example is the inner product in abstract vector spaces. These forms have applications in geometry and physics. In linear algebra, the quadratic form is another common example of a symmetric bilinear form.

5. How are symmetric bilinear forms used in applications?

Symmetric bilinear forms have many applications in mathematics, physics, and engineering. They are used to define important concepts like orthogonality and perpendicularity, and are essential in the study of inner product spaces. In physics, they are used to describe physical quantities like forces and energies. In engineering, they are used in optimization problems, such as finding the minimum or maximum of a function.

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