What is the definition of a field?

[purnapragna]

hi i m very new to this topic as i do not have sound background in group theory

when i was reading book on matrices it was quated that the elements of a matrix are assumed to be scalars from the scalar field F . What does this actually mean.

please note that i m very new to this so do not give great amthematical emphasis on explanation. please give intutive reasoning with some anolgy.

thank you

purna!

 

[steve10]

Re: what is a field?

"Field" is a concept in "Abstract Algebra". Simply put, a "field" is such a set that satisfies:
(1) it has a "zero" and a "unit";
(2) it is closed with respect to addition, subtraction, multiplication and division (except by "zero").

Example 1. The set of all rational numbers is a field.
Example 2. The set of all real numbers is a field.
Example 3. The set of all complex numbers is a field.
Example 4. The set of all nonnegative rational numbers is NOT a field, because it is not closed with respect to subtraction. For instance, 1-2 = -1 which is not in the set.

Example 5. The set of all integers is NOT a field because it is not closed with respect to division. For instance, 1/4=0.25 - not an integer.

Example 6. The set of all matrices is NOT a field because there exist such matrices which are not zero but don't have inverses (not closed with respect to division).

In your case, the entries are assumed to be in a field. I believe it emphasizes that you don't have to use real numbers as the components as usual, and you may want to use other numbers like rational numbers, complex numbers, etc.

 

[jallem]

what is a field?

maybe your next question would be: what does it
mean "closed with respect to..."; means that any
operation(addition, subtraction, multiplication and division) of two members in the field results in another
member belongning to the field.

 

[greenfrog]

Re: what is a field?

a field actually ONLY means a set of pre-defined values, those values in the field satisfying predefined rules. thus to understand a field, u need to know its ranges, and its rule.

when we pick up a value from a field, we said that thsis value is in the field.

scalar means the value has only one component

 

[purnapragna]

Re: what is a field?

then how do you define a random field? What does this mean? does it has same definnition of the field?. Does that mean the numbers taken from this field are random?

thnx

purna!