# Show that $S$ is a real vector space using the standard operations on $\mathbb{R}^3$ (what are “standard operations” on $\mathbb{R}^3$?)

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 $7$?

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

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 $\mathbb{R}^3$.
• @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. – Johanna Nov 6 '14 at 1:41