We’ve now got the general linear group of all invertible linear maps from a vector space to itself. Incidentally this lives inside the endomorphism algebra of all linear transformations from to itself. In fact, in ring-theory terms it’s the group of units of that algebra. So what can we do with it?
One of the biggest uses is to provide representations for other algebraic structures. Let’s say we’ve got some abstract group. It’s a set with some binary operation defined on it, sure, but what does it do? We’ve seen groups acting on sets before, where we interpret a group element as a permutation of an actual collection of elements. Alternatively, an action of a group is a homomorphism from to the group of permutations of some set — .
Another concrete representation of a group is as symmetries of some vector space. That is, we’re interested in homomorphisms . A “representation” of a group is a vector space with such a homomorphism.
In fact, this extends the notion of a group acting on a set. Indeed, for any set we can build the free vector space with a basis vector for each . Given a permutation on we get a linear map defined by setting and extending by linearity.
We thus get a homomorphism from the group of permutations of to . And then if we have a group action on we can promote it to a representation on the vector space . We call such a representation a “permutation representation”.