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

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.

None.

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.

I have no idea at all.

Thanks for your time.

## 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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