Vector span proof

Question

Let $c_1,c_2,c_3,c_4,c_5$ be vectors in $R^4$

I'm trying to show that the set (call it set A) {$c_1,c_2,c_3,c_4,c_5$} spans $R^4$ if and only if the set (say, set B) of vectors

{$c_1+c_2,c_2+c_3,c_3+c_4,c_4+c_5,c_5+c_1$} spans $R^4$

I tried a proof by contradiction to show that there can't exist a vector b that is formed by a linear combination of vectors from B but not from A, but that doesn't seem to be the case. Am I messing something up somewhere/is there something I could be doing differently, possibly with matrices?

Answer 1

Hint:

$ (c_1 + c_2) = (c_1) + (c_2)$

$ (c_1) = \frac {1}{2} \left[ (c_1 + c_2) - (c_2+c_3)+(c_3+c_4) - (c_4+c_5) + (c_5 + c_1)\right]$

Answer 2

You can actually prove something stronger, that A and B always generate the same vector space, by showing that every vector in A can be written as a linear combination of vectors in B, and every vector in B is a linear combination of vectors in A.