Simple Linear Algebra Problem... is the book wrong?
Let V denote the set of ordered pairs of real numbers. If (a, b) and (c, d) are elements of V and x is an element of R, define:
(a, b) + (c, d) = (a+c, bd)
x(a, b) = (xa, b).
Is V a vector space over R? Justify.
Well, it isn't, but the book claims this is due to the fact (which is referred to by (VS4)), that for each element a in V there exists an element b in V such that
a + b = 0.
However, if c = -a, and d = 0, then (a+c, bd) = (0, 0) always. Am I missing something here?