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.

  • 2
    $\begingroup$ 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. $\endgroup$ – Arthur Feb 14 '15 at 18:47

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.