## 5.1, problem 7The following are excerpts from a mail I sent to a student in our class today, about problem 7 in section 5.1: Dear --------, No, there is no mistake in the phrasing of the problem. This is why I assigned the problem. It is illuminating. :-) _ _ Let u, v be in the vector space in problem 7. By definition, u, v are positive real numbers. So their product uv (as real numbers) is again a positive real number. So it is again in the vector space. _ _ __ So (u,v) ----> uv defines a function "+" from V x V to V, as described in class. (Obviously this + has nothing to do with addition of real numbers or addition of positive real numbers....) Example: 3, 5 are in the vector space. To emphasize that we think of them as elements of the v. sp., write them in bold -- _ _ The vector sum 3 + 5 is the real number 15, which of course is positive, so is in the vector space. So we write it as a bold 15, __ 15 , to emphasize that we now think of it as a "vector" in a vector space. And if c is a real scalar and v is a pos. real number, the scalar multiple _ c cv is by definition the positive real number v . So e.g. if c is the perfectly good real scalar -1/2, and v is 7.3 , then _ -1/2 cv is by defn. (7.3) , i.e., 1 over the nonnegative square root of 7.3. (Or rather, it is this positive real number written in bold, since we think of it now as a "vector".) You asked about what notation is ok to use, and what should appear in a proof for problem 7 -- As for notation etc., just be careful with the definitions, maybe use bold (as I suggested in class) where it is helpful, and write down enough to convince the grader that you have *checked* that the operations, as described, do indeed define functions "vector addition" and "scalar multpcn." (I have basically done this for you for free above, where I say that certain things are positive reals.) Then carefully verify that the 8 vector space axioms *do* hold. For this part, you are allowed to use familiar properties of the "field" |R of real numbers. This should be easy. The hard part is "swallowing" the initial definitions, as you have noticed! :-) (In checking the axioms, don't pull the bonehead stunt so many students do, of giving "examples". The axioms are supposed to hold for |

The Chief