# Homework Help: Sub spaces

1. Feb 8, 2010

### EV33

1. The problem statement, all variables and given/known data
If U and V are subsets of R^n, then the set U+V is
defined by

U+V={x:x=u+v,u in U, and v in V} prove that U and V are subspaces of R^n
then the set U+V is a subspace of R^n.
I am just having trouble proving U+V is a subspace.

2. Relevant equations

To be a sub-space...
1. it needs to contain the zero vector
2. x+y is in W whenever x and y are in W.
3. ax is in W whenever x is in W and a is any scalar.

3. The attempt at a solution

1. U and V both contain the zero vector, so their sum will also contain the zero vector.
2. any u1 plus u2 should be in U because U is a subspace, and any v1+v2 should be in V becuse V is a subspace. So (u1+v1)+(u2+v2)=(u1+u2)+(v1+v2)=u+v.
3. below is just the matrix u+v times a
a(v1+u1)=av1+au1
(v2+u2)=av2+au2
(v3+u3)=av3+au3

Because u is in U, and v is in V then au must be in U, and av must b in V,
and u+v is in U+V. Therefore a(U+V)must be in U+V.

Is this sufficient?

2. Feb 8, 2010

### Dick

Yes, it's sufficient. Though I'm having a little trouble understanding why you felt you needed u1, u2, u3 and v1, v2, v3 in part 3. a(u+v)=au+av. Isn't that enough?

3. Feb 8, 2010

### EV33

I did that because I am a linear algebra newb lol.

Thanks for the help.

4. Feb 8, 2010

### Dick

That's a good reason! :)