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**Jameson** Let $\displaystyle {u,w,v}$ be a basis for vector space V. Prove that $\displaystyle {u+v+w, v+w,w}$ 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 $\displaystyle {u,v,w}$.

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 $\displaystyle a_1u+a_2w+a_3v$ can represent any vector in V by definition. Then $\displaystyle a_1u$ can be written in terms of the "basis" we are trying to prove as $\displaystyle a_1(u+v+w)-a_1(v+w)$, $\displaystyle a_2w$ as simply $\displaystyle a_2w$, and $\displaystyle a_3v$ as $\displaystyle a_3(v+w)-a_3w$.

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