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?
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?