Question

what is the difference between a group theory and vector spaces

Answer 1

a group of what?

Answer 2

Group of Vectors?

Answer 3

Vector are made out of objects that have to be vectors. So in a sense a group of vectors and Vector spaces are similar in that way but in order for a group of vectors to be consider a vector space it must follow some specific rules.

Answer 4

A vector space is a group together with some added structure to permit various operations with "vectors" (mathematically, these need not be the magnitude and direction sort but can be other things that satisfy the axioms).