Comment by selimthegrim

I learned from Apostol and Lax, so maybe those are my biases talking, but I don’t remember rank being covered in more than one place at least in the former. Apostol proves it directly from the indices of linear maps. I could see introducing rank in the context of saying in real life, instead of basic Gaussian elimination, we have to consider condition number, then take the reciprocal of the condition number, talk about rank singularity, and then from there touch upon rank (punting to linear maps for the whole treatment, which you did acknowledge as an option) I think a straightforward more real world example oriented to say a basic REPL like that will pique their interest.

Also I hated hated hated the finding reduced row echelon problems in Apostol so maybe it’s just me and I’m lazy.