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?