OIMU 2009 Problem 2

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$ , $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$ . Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.