Unusual Vector Spaces Hi I am a 2nd year undergraduate student studying Mathematics. I have been given the following question on an example sheet and I've been trying to figure it out for a while now and I can't get my head around it.

Consider the triple $(V, \oplus, \odot)$. With $V=\Re$, $u \oplus v = u + 2v$ and $\lambda \odot u=\lambda u$ for any $u, v \in V$ and any $\lambda \in \Re$.

The question is, does $V$ form a vector space over $\Re$? Giving a reason.

I think that you have to show that $V$ satisfies the 8 properties of vector spaces. However the definition of $\oplus$ is confusing me.

When I try to prove the Associativity of addition property - $\forall u, v, w \in V (u \oplus v) \oplus w = u \oplus (v \oplus w)$

I am not sure how to deal with the introduction of $w$ (with $w \in V$).

Does $(u \oplus v) \oplus w = (u + 2v) + 2w$
or does $= (u+2v) + w$? Either way, I can't see how this property can be satisfied

Any help will be greatly appreciated.
Many thanks
Hint: The first one is correct. Now what happens when you change the association by moving the grouping symbols to the pair in the other $\oplus$? For associativity to hold, the results must be equal for all choices of $u,v,w$. Is that the case?
You can't see how this property can be satisfied because with this definition, no, it can't be satisfied. You have just proved that this thing fails to be a vector space. That is how you do it.