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

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In summary, the exercise is asking for a proof that the function [ ]B: L(V) -> Mnxn(R) is an isomorphism, where T -> [T]B is the B-matrix for T, and T is a linear transformation in the vector space of all linear transformations. This means that every matrix and linear transformation have a unique relationship, and the exercise is asking to prove this relationship. The title may have initially mentioned rank, but the focus is on proving the relationship between matrices and linear transformations.
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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...
 
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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'.
 
  • #3
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.
 

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