I have a question regarding vector space, to be more accurate the additive identity axiom. I am used to thinking that additive identity simply means add (0,0,0,...) to a vector and get back the vector. I was however told to not approach a problem that way. How would you guys prove this problem fails to satisfy the additive identity. I think something like $(x_1, y_1)$ + (0,0) = $(x_1, y_1) \neq (x_1 + 0, 0)$ would suffice.
In $\mathbb{R}^2$, consider the following operations:
$(x_1, y_1) \oplus (x_2, y_2) = (x_1 + x_2, 0)$
$\alpha \odot (x,y) = (\alpha * x, y) $
is $\mathbb{R}^2$ with these operations a vector space? list all the vector spaces axioms that fail to be satisfied.