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My textbook states that "A vector space V over a field F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so that for each pair of elements, x, y in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold..."

In our problem set, the professor created a problem where V is over ##\mathbb{R}^+##, but the scalars are members of ##\mathbb{R}##. There is a special definition of addition and scalar multiplication, so I can easily prove that the axioms hold.

My concern is with the use of two different fields. I'm thinking that this is not a vector space because of the field membership.

The issue I'm having with the textbook problem is that a true or false makes the claim "If f is a polynomial of degree #n# and #c# is a nonzero scalar, then #cf# is a polynomial of degree #n#." If I follow the textbooks definition the way I'm interpreting it, I get the answer True. But if I use two different fields, the answer will be false. For example, let the polynomials be from ##Z_3(x)## but the scalars from ##Z_7## and defined multiplication and addition with mod 3. The #c = 6# is a nonzero scalar, but (6 * 2x) mod 3 = 0.

In short, I cannot reconcile what the professor is doing in his question with what the definition appears to say to me. Is there something I'm not seeing?