# Does there exist a linearly independent subset in any linearly dependent set of vectors?

Given any linearly dependent set of vectors $V$, does it hold that there exists a linearly independent subset of $V$, say S?

I can think of examples such as a set of vectors in $\mathbb R^n$ or in $\mathcal P(\mathbb R)$ (the space of all polynomials over $\mathbb R$) where this holds and there are common ways of finding a linearly independent subset. But, does this hold for any set of vectors in any vector space? Any hints or a sketch of a proof would be appreciated.

• Excluding boring examples involving the zero vector and empty subsets, there is always one vector, and if not all vectors in $V$ are parallel, then there is always one more etc. – Arthur Feb 14 '15 at 18:47

## 1 Answer

Clearly, say $V = (v_1, ..., v_n)$, then :

$S = (v_k), v_k \neq 0_E, k \in \{1, ..., n\}$ is a linearly indepentent subset of $V$, so yes there exists at least one linearly indenpendant subset of $V$ if it's not all zero.