Ah, I see. So vectors are treated like abstract objects and representing them in a matrix-like form is an additional step. And instead of coordinate vectors, which may be confusing, you only work with matrices. I can imagine that this is a useful perspective when you work with many different bases. Thank you for sharing it.
Would you then agree to define A⋅v:=A⋅M(E,v) where E is the standard basis?
I wouldn’t be heartbroken if it was defined like that, but I wouldn’t do it if I were writing a textbook myself. I think the LADR approach makes the most sense – vectors and matrices are fundamentally different – and if you want to bring a vector into the matfrix world, then why not demand that you do it explicitly?
If you actually use LA in practice, there is nothing stopping you from writing Av. You can be ‘sloppy’ in practice if you know what you’re doing while thinking that drawing this distinction is a good idea in a theoretical text book.
Ah, I see. So vectors are treated like abstract objects and representing them in a matrix-like form is an additional step. And instead of coordinate vectors, which may be confusing, you only work with matrices. I can imagine that this is a useful perspective when you work with many different bases. Thank you for sharing it.
Would you then agree to define A⋅v:=A⋅M(E,v) where E is the standard basis?
I wouldn’t be heartbroken if it was defined like that, but I wouldn’t do it if I were writing a textbook myself. I think the LADR approach makes the most sense – vectors and matrices are fundamentally different – and if you want to bring a vector into the matfrix world, then why not demand that you do it explicitly?
If you actually use LA in practice, there is nothing stopping you from writing Av. You can be ‘sloppy’ in practice if you know what you’re doing while thinking that drawing this distinction is a good idea in a theoretical text book.