S.O.S. Mathematics CyberBoard

My original solution is "airtight". The only assumptions are those that are given, the v's are dependent, and the w's have the given definitions. I do not need the assumption that the v's are all non-zero, although the case when they are all 0 is vacuous, but the proof is still valid.

Not quite. You wrote

.

That's not the definition of linear dependency, so your proof uses an unverified assumption, so it is invalid.

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What's wrong with the definition?

http://en.wikipedia.org/wiki/Linear_independence

(Quotation from above)
Definition

A finite subset of n vectors, v1, v2, ..., vn, from the vector space V, is linearly dependent if and only if there exists a set of n scalars, a1, a2, ..., an, not all zero, such that

a1v1 + a2v2 + ... + anvn = 0.

Note that the zero on the right is the zero vector, not the number zero.

Yours says "nonzero c's" which means all of them are nonzero, the definition requires only one to be nonzero.

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You are reading it wrong --- "not all zero" is not "all not zero"

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I stand corrected. However the proof itself is valid with the proper wording - in my mind I was always thinking not all zero, even though I reversed "not all" in the text.

Note the final comment: The a's cannot all be 0, since that would lead to all the c's being 0.