Linear algebra

vector space

Group : a non-empty set g with an operation (*) is called a group if the following properties are satisfied. (a) closure law. (b) associative law. (c) existence of identity. (d) commutativity law. examples: (z,+), (Q,+) , (R*,x)

ring : a non empty set r with the operations (x) and (+) respectively denoted by (+,.) is called a ring if the following properties are satisfied.

(a) (R,+) is abelian group (b) for a,b belongs to R implies (ab) belongs to R (closed under multiplication) (C) a,b,c belonging to R (ab)c= a(bc) multiplicative is assosiative. (d) distributive law a(b+c)= ab+ac (a+b)c= ac+bc

then R is ring denoted by (R,+,.)

definition : A non empty set 'F' with operation addition and multiplication denoted by(+,.) respectively is called a field if the following properties are satisfied. (a) (F,+) is abelian group (b) (F*,.) is abelian group (c) distributive properties a(b+c)= ab+ac denoted by (F,+,.) examples: (Q,+,.), (R,+,.), (C,+,.)