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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?

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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.

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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.

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