Hi, I am really having trouble with questions regarding proving whether a given set is a vector space or not.
So one of the questions is [ x ε R2|x12=x23 ]
So I have to prove whether the following set is a vector space
The Attempt at a Solution
Having a peek at the textbook, it tells me that there are ten axioms that satisfy whether a set belongs to a vector space.
What I did was go x12 - x23 = 0
u+ v = v + u -For this one, I did the following: I let x1 and x2 have a value each and named it a vector u, then I let x1 and x2 have some other values and named it a vector v.
So x1 = 1 and x2 = 2 and u = 12 - 23
And x1 = 4 and x2 = 5 and v = 42 - 53
u + v = v + u should work out and I used the same method for the rest of the axioms....I need to know two things
(a) Is the method I am doing correct? If not, could you point me in the right direction?
(b) How do I go about checking the first axiom for the above set u + v lies in the set. If I do the same old substitution, all i get is a number, whatever it is!