TITLE: Difference between generators and basis
QUESTION [5 upvotes]: What is the difference between the terms "generator set" and "basis"?  Don't they both just mean a set of objects that you can use to obtain all of the objects in a larger set under some operations?  Is a basis just a particular kind of generator set or are they completely distinct ideas?

REPLY [5 votes]: In linear algebra a set can generate the space in the sense of linear combinations but if the set is linearly independent then it is a basis. For example in the plane vectors (0,1) and (1,0) constitute a basis because they generate all of plane and are independent while if you consider the set {(1,0),(0,1), v} where v is for example (1,1) then this set generates all of the plane but it's not independent so it's not a basis.