Every finitely generated vector space has a Hamel basis

up:: Finitely Generated Vector Space

Let $(V,+,\cdot )$ be a finitely generated vector space. Then it has, per definition, a finite Spanning Set¹ $S=\{s_1,\dots ,s_m\}\subset V$.

Then we can construct a basis from zero:

• Let $v_1\ne 0$. Then $\{v_1\}$ is Linearly Independent. If it generates $V$, then done.
• If it doesn’t generate $V$, let $v_2\notin [v_1]$, which implies that $\{v_1,v_2\}$ is linearly independent. If it generates $V$, done. Else, repeat.

Since Every linearly independent set of a finitely generated vector space has at most the same number of vectors as its spanning set, this process terminates, at most, with $m$ vectors (the size of $S$).

Thus, a set $B$ constructed in this fashion will be a Hamel Basis, since it’ll span $V$ and also be linearly independent.

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.

Footnotes

1. Note that it need not be linearly independent! ↩