Linear Algebra - question on the proof of the Replacement Theorem in Friedberg

Question

I'm not quite fully understanding the author's use of induction to prove the replacement theorem in their book Linear Algebra:

The induction hypothesis is, as I understand it, that the theorem holds for some integer $m\geq0$. How does the author then assert from this that there exists a set $L=\{v_1,v_2,...,v_{m+1}\}$ with $m+1$ linearly independent vectors? The theorem is assumed to be true for $m$, so to me the set $L$ should have at most $m$ vectors, as there could be no guarantee that the vector space in question has dimension greater than $m$. In fact, what's trying to be proven is precisely that the theorem would also hold for $m+1$, which would then enable such an assertion. What's the logic behind this thought process? Thanks.

Answer 1

This set $L$ is no longer the same set stated in the induction hypothesis (with $m$ linearly independent vectors).

This is $L$ of step $m+1$. In this step, you're trying to prove that given a set $L$ with $m+1$ vectors, you can come up with some set $H$ wherein $\operatorname{span}(LUH)=V$. So, really, he isn't asserting anything (and, consequently, he isn't using circular reasoning); that statement is simply your starting 'condition', which you must show implies the desired result.

Also note that, in this (m + 1)th step, $n$ remains constant. That is, $V$ is still generated by the same set $G$, which has $n$ vectors (just clearing that up).