I am reading the following proof in the book Linear Algebra Done Right. But I don't understand what induction on $j$ means here! I suppose the author means induction on $m$. However, all the proof is talking about $j$ and I don't think this is a typo!
Does induction on $j$ makes sense, here? I mean what is the variable of induction? I am confused!
My Thought
What is the index $j$ that the author is talking about in the proof? Is it the same as mentioned in the theorem? If the author is assuming that the theorem is true for $v_1,...,v_k, 1 \le k \le j$ and then wants to prove it for $v_1,...,v_j,v_{j+1}$ while $j$ remain in $1 \le j \le m-1$, then the proof suffers from abuse of notation so badly.
Induction on $j$ seems meaningless to me! Because theorem is an expression which depends on $m$ (the length of the list of linearly independent vectors) not $j$! In other words, we want to see that Theorem is true for all $m \ge 2$ or not!
$1$- If I was going to prove the theorem by myself then I would change the last line of theorem by
$$\text{span}(v_1,...,v_m)=\text{span}(e_1,...,e_m)$$
and then do an induction on $m$.
$2$- I also think that we can prove the theorem by induction on $m$ in the original form. It is stronger than the case $1$ above?