# Prove that Wm is a subspace of R2?

1. Oct 8, 2011

### krozer

I need help with this problem that I don't know how to solve.

1. The problem statement, all variables and given/known data

For each positive integer m, it's defined a subset of R2 as Wm={(mx,x)|x in R}
(a) Prove that each Wm is a subspace of R2.
(b) ¿Is the union of all Wm a subspace of R2?. Prove it.

2. Relevant equations

None.

3. The attempt at a solution

Trying to prove (a)
To prove Wm is a subspace we know we have to show that
(1) 0 vector is in Wm
(2) Given u,v in Wm; then u+v is an element of Wm
(3) Given u in Wm, k in R, then k*u is an element of Wm

I don't know how to prove (2)
I have a problem proving (3), if we choose k in R, if k is negative, then km is negative and we'd have Wm=((km)x,x) with km negative, then ku for some u in Wm is not an element of Wm.

Trying to prove (b)
I have no idea at all.

Last edited: Oct 8, 2011
2. Oct 8, 2011

### spamiam

To prove (2), you need to write out what u and v "look like" since they are elements of $W_m$. For (3), I think you're making a mistake in your multiplication. Normally the definition is
$$c(x,y) = (cx, cy) \; .$$

For (b), try to think of what $W_m$ represents geometrically. If you're having trouble with it, you can always try drawing an example for a specific value of m.

3. Oct 8, 2011

### krozer

Thanks, proving (2) giving u is an element of Wm then u=(mx,x) x in R, now I know how to prove Wm is a subspace. Proving 3, you're right I forgot about the second coordinate.