Model Answers to the First Homework
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چکیده
1. Chapter 4, §1: 2. Suppose that F is a field and that a and b are in F. Suppose that a · b = 0, and that b = 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0 = 0 · c = (a · b) · c = a · (b · c) = a · 1 = a. Thus a = 0. It follows that F is an integral domain. 8. We compute a b c d 1 0 0 0 = a 0 c 0 , and 1 0 0 0 a b c d = a b 0 0. If we have equality, then b = 0 and c = 0 and vice-versa. Thus the only matrices that commute with 1 0 0 0 are matrices of the form a 0 0 b , where a and b are arbitrary. 10. (a) One way to do this is to write down three general matrices and compute the triple product explicitly, both ways, and check that you get the same answer. A simpler, and possibly more aesthetically pleasing approach, is as follows. Note that a 2 × 2 matrix A determines a map f : R 2 −→ R 2 ,
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