Difference between generators and basis

Question

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?

Answer 1

I think that basis is a particular generator of a set. Unlike any other genrator, a basis conatains the least number of elements that would generate the whole set. Note that anay generator having the same number of elements as the basis, is also a basis.

Answer 2

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.

Answer 3

In linear algebra, a generating set in which all of the elements are linearly independent is called a basis. It's possible to create a linearly dependent set that is still a generating set.