I understand that for a function or a set to be considered a vector space, there are the 10 axioms or rules that it must be able to pass. My problem is that I am unable to discern how exactly we prove these things given that my book lists some weird general examples.
For instance: the set of all third- degree polynomials is not a vector space but the set of all fourth degree or less polynomials is? Is this because I can have $$f(x) = x^3$$ $$g(x) = 1 + x - x^3$$ $$f(x) + g(x) = 1 + x$$ which isn't in 3rd degree where as the fourth degree or less means I can have it not have to be in 4th degree?
Other curious sets I can't seem to discern or wrap my head around include
The set $${(x, y)}$$ where $$x>=0$$ and y is a real number.
A 4x4 matrix with symmetrical ordering except the diagonal is 0, 0, 0, 1 in descending order.
And the set of all 2x2 singular matrices.
The most confusing of all is what I like to call the modified set which changes an operation or two into something like this:
Let V be a set of all positive real numbers; determine whether V is a vector space with the operations shown below
$$x + y = xy$$ $$cx = x^c$$
If anyone could help explain why these sets break whatever axiom (because I just feel like these sets fit as a vector space but my book says otherwise) it would really help me out. I just started the vector space unit in my class and I gotta say being this clueless is scary.