ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4290 on D2043: Set of linearly independent sets
Subresult of R4289: Vector space always has a basis

Vector space always has a linearly independent subset

Formulation 0

Let $R$ be a D273: Division ring.
Let $V$ be a D29: Vector space over $R$ such that

(i)	$\mathcal{L} : = \mathcal{L}_R(V)$ is the D2043: Set of linearly independent sets in $V$ over $R$

Then \begin{equation} \mathcal{L} \neq \emptyset \end{equation}

Proofs

Proof 0

Let $R$ be a D273: Division ring.
Let $V$ be a D29: Vector space over $R$ such that

(i)	$\mathcal{L} : = \mathcal{L}_R(V)$ is the D2043: Set of linearly independent sets in $V$ over $R$

This result is a particular case of R4289: Vector space always has a basis. $\square$