What is the definition of a field?

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!

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.

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.

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

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!