Finitely Generated Vector Space has Finite Basis
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Theorem
Let $K$ be a division ring.
Let $V$ be a finitely generated vector space over $K$.
Proof
This follows from Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set.
It suffices to find:
- A linearly independent subset $L\subset V$
- A finite generator $S\subset V$
with $L\subset S$.
By Empty Set is Linearly Independent, we make take $L = \O$ and $S$ any finite generator, which exists because $V$ is finitely generated.
$\blacksquare$
Also see
- Vector Space has Basis, which does not assume that $V$ is finitely generated
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.8$