So I have this question I'm stuck at at, and it's where I have to prove that some set is not a vector space. First part of the question, I had to prove for 2 different sets with different operations that they're vector spaces, already did that. Now this following set is like this:
Given is set R with the normal addition rules. The Integers (Z) is the Scalar-field and the multiplication of integers and real numbers is defined as the scalar multiplication.
I need to prove that the above is not a vector space. So if I understood correctly, I'm given (R, +) where + is just normal addition so:
r1 + r1 := r1 + r2 for all r1, r1 ∈ R and for the multiplication, it's basically and interger * real number. z * r := z * r for all z ∈ Z and r ∈ R
I've checked all of the axioms for Vector spaces, and they're all holding out... Not sure what I'm missing...
For "+", it's associative, commutative, there is a neutral and inverse element, and group closure is there. There is a multiplicative neutral element as well, e = 1 and "*" and "+" are distributive and associative... And again, there is group closure since an integer times a real number will always give me something back in R.
Any help would be appreciated... I'm still a newbie in Linear Algebra, and I've been working on this for hours now...