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.

  • $\begingroup$ 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. $\endgroup$ – Josue Sep 10 '19 at 18:02
  • $\begingroup$ Regarding my previous comment, I was having a wrong interpretation, so ignore that. $\endgroup$ – KNilesh Sep 10 '19 at 18:03
  • $\begingroup$ No worries, thank you for clarifying $\endgroup$ – Josue Sep 10 '19 at 18:04
  • $\begingroup$ 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. $\endgroup$ – KNilesh Sep 10 '19 at 18:10
  • $\begingroup$ So no identity and inverse!! $\endgroup$ – KNilesh Sep 10 '19 at 18:14

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.

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  • $\begingroup$ would it then be correct to just assume them working or are there cases in which they fail to be satisfied $\endgroup$ – Josue Sep 10 '19 at 18:23
  • 2
    $\begingroup$ @Josue: No, it's never okay to "just assume them working." You should work out both sides. Either you will get the same answer in both (or answers that are always equivalent), and then you can conclude (not assume) that the equality holds. Or you will get expressions that may fail to be equal, in which case you should exhibit specific values for which they are not equal, in order to conclude (again, not assume) that the equality does not always hold. $\endgroup$ – Arturo Magidin Sep 10 '19 at 18:27
  • $\begingroup$ Understood, thank you. $\endgroup$ – Josue Sep 10 '19 at 18:31

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