The book "A First Course in Algebra" says
In a finite dimensional vector space, every finite set of vectors spanning the space contains a subset that is a basis.
All that is fine. But what about a span having an infinite number of vectors? Surely that too must contain the basis!! An example is $\{\overline{i},\overline{j},\overline{k}+r\overline{i}\},\forall r\in\Bbb{R}$. Can we select all linearly independent vectors in this infinite span, and then prove it is the basis? Is this a sound mathematical technique? The proof given for finite spans does not seem to suggest this.
Thanks in advance!
Best Answer
Given that $V$ is finite-dimensional one can argue as follows: Take a basis $(e_i)_{1\leq i\leq n}$ of $V$. Each of the $e_i$ is a finite linear combination of vectors from the spanning set $S$. It follows that there is a finite subset $S_0\subset S$ that spans $V$. From your "First Course in Algebra" you can then conclude that $S_0$ contains a basis, and so does $S\supset S_0$.