Linear transformations and subspaces

[Deneb Cyg]

Homework Statement

Let B={b1,b2} be a basis for R2 and let T be the linear transformation R2 to R2 such that T(b1)=2b1+b2 and T(b2)=b2. Find the matrix of T relative to the basis B.

The Attempt at a Solution

I know that the matrix I'm looking for needs to be 2x2 and that the standard matrix of a linear transformation is related to how the transformation would affect the identity matrix. However I don't understand how to relate it to the basis.

My best guess is:
T(b1)=2b1+b2
T(b2)=0b1+b2
so the matrix is [[2,0][1,1]]

But I'm not sure if this is right (if it is, I'm not sure why) or how to check it. Am I on the right track?

 

[matt grime]

You know that the linear map sends (1,0) to (2,1) and (0,1) to (0,1) (read as columns). So all you need to do is check that your matrix does that, and in checking it you should see why you did what you did to get the answer.

 

[Deneb Cyg]

Okay, that makes sense. The second part of the question is "suppose now that b1=(1,1) and b2=(1,2) (read as columns). Find the matrix of T relative to the standard basis of R2."

Would the answer for this be the same [[2,0][1,1]] as before because its the same transformation or does defining B and relating it to the standard basis instead change the answer?

 

[matt grime]

Changing basis changes T. I assume that when b1=(1,1) this is b1 relative to the standard basis e1 and e2 (well, 'standard' doesn't matter - it's just another basis).

There are several ways to do this. You could just try to work out what Te1 and Te2 are from the action of T on b1 and b2 (e.g. note that e2=b2-b1, so you know what Te2 is now, but don't forget to express Te2 in terms of e1 and e2.).