I have the following question regarding three axioms of vector spaces.
consider the following case.
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.
how can I prove
1) Associativity of scalar multiplication
2) Distributivity of scalar sums
3) Distributivity of vector sums.
As of right now, In all the problems I had solved I always assumed that these three axioms were real. I assumed that because if the commutative and associative additive axioms were satisfied then these three axioms would be a direct result of them. Is my thought correct? Also would anyone have an example in which one of these axioms fails to be satisfied.
To solve this type of problem you set up the identity required then check to ensure both sides of the equality match or don't match.
For example consider the associativity of scalar multiplication which states that $\alpha \odot (\beta \odot v)=(\alpha * \beta )\odot v$. You can then expand each side using the operations given to see if they are the same or not being careful to evaluate them exactly as written. Note that all the axioms you're asked to verify are of this form with two different equations on either side of an equal sign so it's the same idea to solve all of them. Most of the work is notational bookkeeping.