1
$\begingroup$

Consider the following claim: Let $V$ be a vector space and let $A,B\subseteq V$ be two independent sets with $|A|<|B|<\infty $. Then there exists $b\in B$ such that $A\cup \{b\}$ is independent.

Can anyone prove this claim without using matrices?

  • 1
    Something like [this](http://en.wikipedia.org/wiki/Steinitz_exchange_lemma) might help.2012-06-13
  • 0
    The basic result here is the (Steinitz) *Exchange Lemma*. A web search should turn up much, including that it generalizes from *linear* dependence to *abstract dependence relations* (e.g. *algebraic* dependence), cf. *matroid* theory.2012-06-13

3 Answers 3