From Elementary Algebra by Howard Anton 9th Edition, Problem 16: From Elementary

Question

From Elementary Algebra by Howard Anton 9th Edition, Problem 16: From Elementary Algebra by Howard Anton 9th Edition, Problem 16: A set of objects is given, together with operations of addition and scalar multiplication. Determine whether the set is a vector space under the given conditions. If the given set isn't a vector space, list the axioms which fail to hold. Here are the two axioms I'm using for this question: Problem 16: (x,y)+(xprime,yprime)= (x(xprime),y(yprime)) and k(x,y)=(kx,ky) There's some discrepancy in the solution to this problem. To me, axiom 4 should hold since if u=(x,y) , 0=(1,1). Now, whether the set agrees with axiom 5, I'm not sure. -u should equal (1/x,1/y) so that u+ (-u) = 0.   However, if x or y = the number 0 in this case, there is no -u due to division by the number 0 since the set should include "all pairs of real numbers" (number 0 included). Therefore, I would conclude that axiom 5 fails. Is this correct?
Thank you.

Explanation / Answer

You're right. Actually for this set to be a vector space, we should not include the element (0,0). Note also that (0,0)+(5,5) = (0,0) and (0,0)+(1,1) = (0,0) But axiom 4 says (5,5) is not a zero vector, right? since (x,y)+(5,5) =/ (not equal) (x,y) The (0,0), (0,y), (x,0) are the culprits in this set. I think there are other axioms violated by them. P.S. I hope this helps. In group theory, there are stuffs called "generators" of the group. Think of it, is there any number aside from zero in which you can do the operation and get 0?

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