# Vector Space Axioms intuition

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.

• sorry would you care to elaborate. I am aware that I fail to satisfy the scalar multiplication identity if that is what you are referring to, but how can I relate that to prove the three axioms I mentioned. – Josue Sep 10 '19 at 18:02
• Regarding my previous comment, I was having a wrong interpretation, so ignore that. – KNilesh Sep 10 '19 at 18:03
• No worries, thank you for clarifying – Josue Sep 10 '19 at 18:04
• about these three axioms, you are correct that if commutative and associative axioms hold for addition which is true definitely for Real number (so holds here) then these three axioms are satisfied. – KNilesh Sep 10 '19 at 18:10
• So no identity and inverse!! – KNilesh Sep 10 '19 at 18:14

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.