1
$\begingroup$

My problem is:

Let $S = \{(x,y,0):x,y \in R\}$. Show that $S$ is a real vector space using the standard operations on $\mathbb{R}^3$.

What exactly are the standard operations on $\mathbb{R}^3$? I'm not sure if it means closed under addition/scalar mult, or something else added to those, or anything else entirely.. I looked for a while trying to find out what standard operations for showing S is a real vector space is but nothing seems to tell me what standard is specifically..

edit: can someone tell me how to turn the E into the symbol meaning "is in"?

edit 2: taken from last comment in the answer, just to be sure, if I show all 8 axioms hold, I don't have to also show that the set is "closed under vector addition", "closed under scalar multiplication", and "has the 0 vector" separately, do I? Actually I believe those are the part of examples that say "together with these two operations, show ... is a real vector space". I believe "these two operations" refer to closed under vector addition and scalar multiplication, and that the "standard operations on R3" are the 8 axioms. Am I right to assume this?

also, to verify my method of proving; while showing the 8 axioms hold, is it correct to let $u = (x,y,0)$, then show $u+v=v+u$ etc. by writing $u+v = (x,y,0)+(v_1,v_2,v_3)$ ..... then finishing off the proof, along with the other $7$?

$\endgroup$
  • 3
    $\begingroup$ Appendectomy, hemorrhoidectomy, tonsillectomy are the most common. $\endgroup$ – copper.hat Oct 30 '14 at 22:52
  • 1
    $\begingroup$ You could show that $S = \ker \phi$, where $\phi((x,y,z)) = z$. $\endgroup$ – copper.hat Oct 30 '14 at 22:52
  • $\begingroup$ @copper.hat This is the most elegant way. Very nice! $\endgroup$ – brick Oct 30 '14 at 22:59
  • $\begingroup$ I havent learned kernals yet btw. $\endgroup$ – J L Oct 30 '14 at 23:03
1
$\begingroup$

You would need to show that all the axioms for a vector space hold, using normal vector addition and scalar multiplication as you normally would on elements in $\mathbb{R}^3$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ so do I have to apply each of the eight conditions such as 1: addition is communtative, 2: addition is associative.... etc. ? $\endgroup$ – J L Oct 30 '14 at 22:53
  • 1
    $\begingroup$ @JL Exactly, you just have to check all of them. $\endgroup$ – Johanna Oct 30 '14 at 22:54
  • 1
    $\begingroup$ @JL If you have learned about subspaces, then you can just do the subspace test. If you haven't, then you need to write out all the steps. $\endgroup$ – Johanna Oct 30 '14 at 23:05
  • 1
    $\begingroup$ @JL Not quite. "closed under vector addition" and so on ARE some of the axioms. You need to show either that all eight axioms hold, using standard vector addition / scalar multiplication, or show that the space passes the subspace test. Nothing else. $\endgroup$ – Johanna Nov 5 '14 at 2:12
  • 1
    $\begingroup$ @JL I'm not sure which is your first axiom, but to for example show that $S$ is closed under addition, take two vectors $(a,b,0),(c,d,0) \in S$. Then $(a,b,0)+(c,d,0) = (a+c, b+d, o) \in S$, so $S$ is closed under addition. $\endgroup$ – Johanna Nov 6 '14 at 1:41
0
$\begingroup$

edit 1: To get the symbol "is in" type "\in"

This is the best place to start to learn latex imo. http://www.thestudentroom.co.uk/wiki/LaTex

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.