Let

be a basis for vector space V. Prove that

is also a basis for vector space V

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Ok so the show that this is a basis, it must satisfy being linearly independent and must be able to span any vector spanned by the

.

So I can see that the vector sum u+v+w cannot be linearly dependent because of the fact that any vector can be written as a sum of these three vectors and at least one non-zero coefficient from {u,v,w} being a basis. It also follows in the same manner the v+w and w are not linear dependent sums. The problem I have is showing that the sum of all three of these is linearly independent. Hmmm...

As for spanning vector space V, let's say that

can represent any vector in V by definition. Then

can be written in terms of the "basis" we are trying to prove as

,

as simply

, and

as

.

Look ok minus the missing part which I need help on?