Proving Vector Subspace: V in W iff V+W in W

[Juggler123]

I have been given that V is a finite dimensional vector space over a field F and that W is a subspace of V. I need to show that v is an element of W if and only if v+w is an element of W.

I know that because it is an 'if and only if' proof it needs to proved in both directions but don't really know where else to go from there! Any help would be great, thanks.

 

[Dosmascerveza]

I am going to assume that w is given to be in W.

You could show that

If W is a subspace of V then
let w E W be a vector in W. (non empty)
let v E V be a vector in V. (non empty)

if w+v is NOT in W then v is NOT in W due to closure condition. if v is in W then v+w MUST be in W and V.

so you have shown that w+v either is or is not in W and if it is in W then
it must also be in V and if w+v is in W then both w and v must be in W. (closure)

Even If I worded it incorrectly I am sure that this involves closure condition for subspace.

 

[Dosmascerveza]

notice i used this method

A ==> B, not B ==> not A