Let S: R2 --> R2 be the linear map S(x, y) = (y, x). With respect to the basis ((1, 2), (0, 1)) for R2, we have S(1, 2) = (2, 1) = 2(1, 2) - 3(0, 1) and S(0, 1) = (1, 0) = 1(1, 2) - 2(0, 1).  I understand how they got S(1, 2) = (2, 1), but not how they got 2(1, 2) - 3(0, 1) and I also understand how they got S(0, 1) = (1, 0) but not how they got 1(1, 2) - 2(0, 1). Can you show what the steps are to get these?

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