Change of Basis wrt Linear Transformation

Hi, I have one part of a two-part question related to a linear transformation.

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(a) Let a linear transformation T: R2 --> R3 be given by T[1,3]=[2,3,4] and T[-2,5]=[1,0,2]. Find the matrix of T relative to the standard basis in each space.

I actually got this part. You just have to find how T operates on [x,y] by using the fact that T is linear.

From there, it was easy to show that T[1,0] = [7/11,15/11,14/11] and that T[0,1] = [5/11,6/11,10/11].

So, the matrix of T relative to the standard bases is [ T[1,0] ; T[0,1] ].

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(b) Find a basis B of R2 and a basis C of R3 such that T relative to B,C =

[1 0]

[0 1]

[0 0]

I have a notion of what to do, but don't know how to set it up. The image under this matrix would be {[x,y,z]: z=0}, as opposed to a different planar subspace of R3 as in part (a). Do I use a change of basis procedure here? And how would I set it up?

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