General Linear group I am having trouble understanding what a general linear group is? So, given a normed linear space E, what is GL(E)? Of what I have read, it the set of invertible matrices, but I am unclear as to what that means or the significance of it. Also if you have a map $f: A \to GL(E)$, what is the map f represent? 
 A: $GL(E)$ is the group of (bounded) linear isomorphisms of $E$ with itself.  For the spaces you're used to like $\mathbb{R}^n$, examples of elements of this group might look like dilations (stretching the space), rotations, reflections, etc.  If $E$ has finite dimension $n$, then we can choose a basis for $E$, and then the linear transformations of $E$ are naturally in one-to-one correspondence with $n\times n$ matrices.  So for these spaces, it's sometimes convenient to think of the group of invertible $n\times n$ matrices instead of the more 'abstract' group $GL(E)$, since they're pretty much the same object (they're isomorphic groups).
We really like the group of invertible $n\times n$ matrices.  It's a well-understood group that shows up all of the time.  So sometimes, to study an arbitrary group $A$, we might want to represent it as a more familiar group of matrices, and then we can study those matrices instead.  We can do this by taking maps like the one you've described, $f: A\rightarrow GL(E)$, which, if $f$ is a group homomorphism, is called a representation of $A$ on $E$.  It just gives us a new way of trying to study $A$ (and $E$).  Check out the definitions and examples here for more info.
