Linear Algebra: How to prove that given set creates vector space V?

Question

Suppose $(v_1, v_2, ...,v_n)$ creates vector space V. Prove that the below set also creates $V$. $$(v_1-v_2, v_2-v_3, ..., v_{n-1} - v_n,v_n)$$

Any hint how can i start the proof?

Answer 1

Suppose that $n=3$. If $w$ is a vector from your space, then there are numbers $\alpha,\beta,\gamma$ such that$$w=\alpha v_1+\beta v_2+\gamma v_3.$$But then$$w=\alpha(v_1-v_2)+(\beta+\alpha)(v_2-v_3)+(\gamma+\beta+\alpha)v_3.$$Can you do the general case now?