Direct sum of trivial subspaces

Hi guys.

i need help to prove the following:

"Let be subspace of , such that there exist one and only one subspace for wich .

Prove that must be a trivial subspace."

this is what i've tried so far:

let's suppose there exist only one subspace .

and let's suppose it is **not** trivial.

let be the base for .

let be the base for .

let ,

since , there exist one and only one linear combination such that:

according to the assumption, there exist only one , but there are infinite number of bases for .

therefor, i can define new base for W and use other scalares...

now, i'm not sure if i'm in the right way or where, if at all, lies the contradiction.

i could really use some direction here.

thanks in advanced!

Re: Direct sum of trivial subspaces

Suppose and .

let be a base for and a basis for . Set .

You can easily show, I hope, that and

Re: Direct sum of trivial subspaces

Hi johng, thanks for your help.

don't i need to show first that is linear independent?

Re: Direct sum of trivial subspaces

Yes, that's part of the proof.

Suppose , i.e., . Since are part of a basis of , they are linearly independent. Hence all a's are 0 and so are linearly independent.