# Show that an infinite set of families are linearly independent

I'm having problems with the following problem:

Let $$V$$ be a vector space over the field $$F$$. Let $$n \in \mathbb{N}$$. Suppose that $$\left( v_1, \ldots,v_n \right)$$ is a linearly independent family of vectors in $$V$$. Show that the families $$\left( {{v_1},{v_1} + {v_2}} \right),\left( {{v_1},{v_1} + {v_2},{v_1} + {v_2} + {v_3}} \right),$$ $$\ldots,$$ $$\left( {{v_1},{v_1} + {v_2}, \ldots ,{v_1} + {v_2} + \ldots + {v_n}} \right)$$ are linearly independent.

This is my attempt:

I take a typical finite subset such as $$\left( {{v_1},{v_1} + {v_2}, \ldots ,{v_1} + {v_2} + \ldots + {v_n}} \right)$$, where $$n \in \mathbb{N}$$. Then suppose that $$\lambda_1{v_1}+\lambda_2\left({v_1} + {v_2}\right)+ \ldots +\lambda_n \left({v_1} + {v_2} + \ldots + {v_n}\right)=0$$, where $$n \in \mathbb{N}$$.

I know there is the trivial solution $$\lambda_1=\lambda_2=\ldots=\lambda_n=0$$, but I'm not sure how to check if there are others?

The equation $$\lambda_1 v_1+\lambda_2 (v_1+v_2)+\dots+\lambda_n (v_1+\dots+v_n)=0$$ can be written as $$(\lambda_1+\dots+\lambda_n)v_1+\dots+\lambda_n v_n=0,$$ which implies $$\lambda_1+\dots+\lambda_n=0,\quad...,\quad\lambda_{n-1}+\lambda_n=0,\quad\lambda_n=0$$ because $$(v_1,\dots,v_n)$$ is linearly independent, which obviously implies $$\lambda_1=\dots=\lambda_n=0$$ by recursion.
You can rewrite $$\lambda_1 v_1 + \lambda_2 (v_1 + v_2) + \cdots + \lambda_n (v_1 + \cdots + v_n) = 0$$ as $$(\lambda_1 + \cdots + \lambda_n) v_1 + (\lambda_2 + \cdots + \lambda_n) v_2 + \cdots + \lambda_n v_n = 0.$$ Now use the fact that $$(v_1, \ldots, v_n)$$ is a linearly independent set to figure out what $$\lambda_1, \ldots, \lambda_n$$ could be.