Basically the transformation takes the vector X and turns it into B. Since it is a linear there exists a matrix A such that f(X) = AX = B. To get you started on the first question consider that e1 = <1,0,0>, e2 = <0,1,0> and e3 = <0,0,1>.
I would suggest that you find the matrix that takes X and creates B via the matrix A. Just match up the components for every element of B by putting them in the right positions for the matrix A and it should all make sense.