Fields are very important to the study of linear algebra. This is because any result in linear algebra applies to all fields, because the basic operations in linear algebra involve only addition, subtraction, multiplication, and division.
However, they are primarily the study of Abstract Algebra and will not fully be treated here. Instead, we provide only the definition.
A field is a set F with two binary operators (or functions) + and * and with elements 0 and 1 such that:
- Commutativity of addition: a+b=b+a
- Associativity of addition: (a+b)+c=a+(b+c)
- Additive identity: 0+a=a+0=a
- Additive inverse: There exists an element b such for all a such that a+b=0
- Commutativity of multiplication: ab=ba
- Associativity of multiplication: (ab)c=a(bc)
- Multiplicative identity: 1a=a1=a
- Multiplicative inverse: There exists an element b for all nonzero a such that ab=1
- Distributivity of multiplication over addition: a(b+c)=ab+ac
Examples of fields:
- The rational numbers Q
- The real numbers R
- The complex numbers C
- The set of rational polynomial functions
Another important assumption for linear algebra is that we are working with a field of characteristic 0.
The characteristic of a field is the first natural number n such that 1+1+1+...+1 (n times) is equal to 0. If there is no such number, then it is of characteristic 0.
- The integers modulo p, Zp where p is a prime number.
In linear algebra, we do not work with such fields as Zp, so we will only work with fields of characteristic 0.