What's the difference between "generate" and "linear span" in linear algebra? In this question, the word "generate" is used when the author of the book mentioned a vector space. And also, in a Gowers's article, it is used in "vector space generated by the functions $[v,w]$". There is another similar concept called "linear span" in linear algebra. In Gowers's article, he also uses the word "span" as "the space spanned by the functions [v,w]".
Here is my question:

Are these two concepts the same? Or what's the difference?

 A: "Span" is less ambiguous and "generate" is more general. 
In the contexts of vectors in a vector space, "generated by" and "spanned by" mean the same thing. However, "generate" can mean various other things (for example generation as a module, as an algebra, as a field, etc.) if there is extra structure on the vector space. 
A general definition of generation is as follows: let $e_i$ be elements of some structured set $A$. A subset $B \subset A$ is said to be the structure generated by the $e_i$ if it is the intersection of all substructures of $A$ containing the $e_i$. (The key word here is "substructure": the notion of generation changes depending on what kind of structure one is considering on $A$.) Of course this definition only makes sense if the intersection of substructures of $A$ is another substructure, but this is in my experience true of structures to which the word "generate" are applied. 
A: Actually, the intersections of substructures of the same type should be a substructure of that same type. If i've not misunderstood the content, which is not unlikely, this should be the theorem which proves it (literal quote from "Algebra, Vol 1 - L. Redei - Pergamon Press, 1967 pag 74"):
"Theorem 46. Let S denote an A-structure and Sl, S2,... denote B-sub-
structures of S where A, B is a suitable pair of systems of structure axioms
for semigroups, groups, modules, rings or skew fields. If the intersection
D = S1 inters. S2 inters. ...
is not empty, this is also a B-substructure of S, except in the case when A denotes
the ring axioms, B the skew field axioms and D is the zero element of S."
