Prove that one span is equal to another

Question

I have question according to one of my linear algebra final exam course, i don't remember how task was written, but it was about proving that: $span(v_1,v_2,...,v_n) = span(v_1,v_1+v_2,v_2+v_3,...,v_{n-1}+v_n)$
vectors form some same vector space, v1,...,vn are linearly independent

My idea was to prove that second set of vectors is linear combination of first set, since span of linear combination of a set of vectors is equal to span of this set.

Answer 1

Hint In general, you can use the following lemma

Lemma Let $\{w_1,\cdots w_k\}\subset H$ ($H$ is a subspace) then $$ span\{w_1,\cdots w_k\}\subset H $$

here the spans are subspaces. So, can you prove that each $v_i$ is in $$ span(v_1,v_1+v_2,v_2+v_3,...,v_{n-1}+v_n)\ ? $$ Converse is easy.

Note Linear independence is not needed.