Why is that? Note that when I mention the dimension theorem, I'm actually referring to the fundamental rank-nullity theorem.
To be more clear, what I was suggesting is to take the section on rank, toss that into the section on matrices, and then toss that into the section on Gaussian elimination (preceding vector spaces). That's useful because the rank of a coefficient matrix tells you the nature of the solution space of the corresponding system.
Then when you cover the rank-nullity theorem in the chapter on linear maps, you'll have the context of 1) what rank means in a matrix, and 2) what rank means in a linear map. That sets up a deeper understanding of how to verify the rank-nullity theorem for any linear map by finding the rank and the nullity using the basic and free variables of the reduced row echelon matrix.
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.
Why is that? Note that when I mention the dimension theorem, I'm actually referring to the fundamental rank-nullity theorem.
To be more clear, what I was suggesting is to take the section on rank, toss that into the section on matrices, and then toss that into the section on Gaussian elimination (preceding vector spaces). That's useful because the rank of a coefficient matrix tells you the nature of the solution space of the corresponding system.
Then when you cover the rank-nullity theorem in the chapter on linear maps, you'll have the context of 1) what rank means in a matrix, and 2) what rank means in a linear map. That sets up a deeper understanding of how to verify the rank-nullity theorem for any linear map by finding the rank and the nullity using the basic and free variables of the reduced row echelon matrix.
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.