If v1 + w, v2 + w,..., vn + w are linearly dependant, prove w belongs to the span of v1, v2,...,vn

Question

If v1 + w, v2 + w,..., vn + w are linearly dependant, prove w belongs to the span of v1, v2,...,vn

I'm told that this proof is correct, but I believe it is not. I believe it is false because:

If v1 = v2 = ... = vn = 0:

v1 + w, v2 + w,..., vn + w simplifies to w, w,..., w which is linearly dependant, so the first condition is still satisfied.

In this case span(v1, v2, vn) is the 0 vector. Therefore w will only belong to the span of v1, v2, vn if w = 0.

Therefore the proof does not hold true for when v1 = v2 = ... = vn = 0, w != 0

Am I correct in believing the proof is incorrect? Or is it actually correct as I am told it is.

Answer 1

You are right, the assertion as you've written it is not true, and you have found a counterexample. I'm not sure what proof you are looking at, but if the proof claims that $w$ is some kind of linear combination of the $v_i$'s, it should check that there's no division by zero when working out the coefficients of that claimed linear combination.

Answer 2

Your counterexample would be correct. But I think the statement of your question may be wrong. It should be: "Suppose that $\{v_1,v_2,\dots,v_n\}$ is linearly independent. If the set $\{v_1+w,v_2+w,\dots,v_n+w\}$ is linearly dependent, then $w\in span\{v_1,\dots,v_n\}$.

In your "example", the vector $\vec{0}$ is not linearly independent.