TITLE: Generator of End(V)
QUESTION [5 upvotes]: If V is a finite-dimensional vector space of dimension n and G⊂End(V) such that G generates End(V) meaning that any element of End(V) is expressible as a linear combination of products of a number of elements of G, what is the minimal possible number of elements in G?

REPLY [4 votes]: Burnside's irreducibility criterion states that an action of a group on a vector space (over an algebraically closed field) is irreducible if and only if the image generates $End(V),$ as you describe. Now, since every finite simple group is generated by two elements, the answer is $2,$ at least for an algebraically closed field.
In fact, if you look at this Mathematica file you will find small generating sets of $SL(n, \mathbb{Z})$ and $Sp(n, \mathbb{Z}).$ Since the two groups obviously act irreducibly, you have (algebra) generating sets of $M^{n\times n}$ with two or three elements.