Every finitely generated vector space has a Hamel basis

up:: Finitely Generated Vector Space

Let be a finitely generated vector space. Then it has, per definition, a finite Spanning Set¹ .

Then we can construct a basis from zero:

• Let . Then is Linearly Independent. If it generates , then done.
• If it doesn’t generate , let , which implies that is linearly independent. If it generates , 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 vectors (the size of ).

Thus, a set constructed in this fashion will be a Hamel Basis, since it’ll span 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! ↩