Linear dependence and relations

#linear_algebra

Suppose $V$ is a vector space, and $v_{1}, v_{2}, \dots, v_{n}$ are in $V$ . Prove that either $v_{1}, \dots, v_{n}$ are linearly independent, or there exists a number $k \leq n$ such that $v_{k}$ is a linear combination of $v_{1}, \dots, v_{k - 1}$ .

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Suppose $V$ is a vector space, and ${\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n$ are in $V$. Prove that either ${\bf v}_1, \ldots, {\bf v}_n$ are linearly independent, or there exists a number $k\leq n$ such that ${\bf v}_k$ is a linear combination of ${\bf v}_1,\ldots, {\bf v}_{k-1}$.