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Prove that Wm is a subspace of R2?

  1. Oct 8, 2011 #1
    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


    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.

    Thanks for your time.
    Last edited: Oct 8, 2011
  2. jcsd
  3. Oct 8, 2011 #2
    To prove (2), you need to write out what u and v "look like" since they are elements of [itex] W_m [/itex]. 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 [itex] W_m[/itex] represents geometrically. If you're having trouble with it, you can always try drawing an example for a specific value of m.
  4. Oct 8, 2011 #3

    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.
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