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A question on my book says the following: Prove that any three vectors u,v and w are linearly dependent.

I don't see how this could work, if the vectors could be any three vectors, we can take into consideration the case where we assume that one or more is 0.

If one of the three vectors is 0, then the statement could be rephrased as: prove that any two vectors are linearly dependent. This is an impossibility, so the original statement must also be impossible to prove.

Am I right in thinking this or am I missing something?

Victoriankoifly
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  • Please tell us exactly why you think it is 'legitimate', and why your statement can be rephrased as you did. (Lets say $u=v=w=0$. Then any two of these vectors are linearly dependent.) Please clarify your use of any, see http://math.stackexchange.com/a/402073/109451 – flawr Jan 15 '15 at 19:45
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    Linear dependence usually involves non zero vectors. Since the presence of the zero vector in a set makes that set always linearly dependent. – Kaster Jan 15 '15 at 19:45
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    If fact if any of the vectors is $0$, then the set is linearly dependent. – Tim Raczkowski Jan 15 '15 at 19:45

1 Answers1

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The statement is of the form $\forall u,v,w\in\mathbb{R}^2$, $u,v$ and $w$ are linearly dependent. That means that no matter what three vectors you choose, they must be linearly dependent. Of course $0$ might be one of the three vectors chosen, but the statement is much more general than that, since it is not essential that one of the three vectors be $0$.

To prove the statement, you could form the $3\times2$ matrix, call it $A$, with rows $u,v$ and $w$. The rank of $A$ shall be $\leq2$ which is $<3$, therefore $u,v,w$ are linearly dependent.

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