(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Sub spaces

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