Prove that Wm is a subspace of R2?

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In summary: It's easy to prove. For (b) Wm are lines through the origin, so the union of all Wm is the set of all points on the x-axis. This set is not closed under addition, so it is not a subspace of R2.In summary, the conversation discusses proving that each subset Wm of R2 is a subspace, and whether the union of all Wm is also a subspace. It is proven that Wm is a subspace by showing that it satisfies the three conditions for being a subspace. However, it is also proven that the union of all Wm is not a subspace because it is not closed under addition.
  • #1
krozer
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I need help with this problem that I don't know how to solve.

Homework Statement



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.

Homework Equations



None.

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.
 
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  • #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
[tex]
c(x,y) = (cx, cy) \; .
[/tex]

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.
 
  • #3
spamiam said:
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
[tex]
c(x,y) = (cx, cy) \; .
[/tex]

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.


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.
 

1. What is a subspace?

A subspace is a subset of a vector space that follows all of the same rules and operations as the larger vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

2. How can we prove that Wm is a subspace of R2?

In order to prove that Wm is a subspace of R2, we must show that it follows the three defining properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector.

3. What is the zero vector in Wm?

The zero vector in Wm is the vector (0,0). This is because the zero vector must be contained in the subspace, and all vectors in Wm have a zero second component.

4. How do we show closure under addition in Wm?

To show closure under addition, we must take two arbitrary vectors in Wm and show that their sum is also in Wm. This can be done by showing that the sum of the first components equals the sum of the second components, as this is the defining property of Wm.

5. Can Wm be a subspace of any other vector space besides R2?

No, Wm can only be a subspace of R2. This is because the definition of Wm requires that all vectors in the subspace have a second component of zero, which is specific to R2.

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