Proving the Isomorphism of [ ]B: L(V) to Mnxn(R) in Linear Transformations

[sweetiepi]

Prove that the function [ ]B: L(V) -> Mnxn(R) given by T -> [T]B is an isomorphism. [T]B is the B-matrix for T, where T is in the vector space of all linear transformations.

I don't quite understand this...

 

[Landau]

Please elaborate: what don't you understand? Do you know what an isomorphism is?

What this exercise says, is that given a basis, every matrix determines a unique linear transformation, and vice versa. You probably already knew this, now you have to prove it. It really is "writing out the obvious".

PS: I don't quite see the connection with 'rank'.

 

[sweetiepi]

Oh sorry about the confusion with the title. When I initially posted a question, it had to do with rank, but then I figured that one out so I just edited the question rather than post a new thread and must have forgotten to change the title.

I suppose my confusion was just because it's hard for me to visualize what's happening with coordinate mappings for a B-matrix for a transformation. I tend to have the most difficulty with the obvious things. But regardless, I went to talk to a GSI about it today and he actually helped a lot. I just needed help getting the problem started, but I got it now.