Any linearly independent set can be extended to a Hamel basis

up:: Linear Independence

Given a linearly independent set $S\subset V$ from a Vector Space $(V,+,\cdot )$, we can extend it to a Hamel Basis by aggregating vectors outside of its Span¹ ─ which guarantees to terminate at the same number of elements of any basis, since Every linearly independent set of a finitely generated vector space has at most the same number of vectors as its spanning set ─ and, in particular, of Hamel bases.

Thus, this augmented linearly independent set will also be a spanning set ─ thus, a basis.

Footnotes

1. Since The union of a linearly independent set with a vector outside of its span is also linearly independent. ↩