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

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.

• Similar if not same doubt was raised in this question and was resolved by OP. Aug 17, 2022 at 5:31

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).