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

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.

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  • $\begingroup$ Similar if not same doubt was raised in this question and was resolved by OP. $\endgroup$
    – Kurt G.
    Aug 17, 2022 at 5:31

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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).

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