Just to share my two cents on the matter, the distinction between abstract vectors and maps on the one hand, and columns with numbers in them (confusingly also called vectors) and matrices on the other hand, is a central headache for Linear Algebra students across the globe (and by extension also for the lecturers). If the approach this book takes works for you then that’s great to hear, but I’m wary of `hacks’ like this that only supply a partial view of the distinction. In particular matrix-vector mulitplication is something that’s used almost everywhere, if you need several translation steps to make use of this that could be a serious obstacle. Also the base map ΦB:Fn→V that limerott mentions is of central importance from a category-theoretic point of view and is essential in certain more advanced fields, for example in differential geometry. I’m therefore not too keen on leaving it out of a Linear Algebra introduction.
Unfortunately I don’t really know what to do about this, like I said this topic has always caused major confusion and the trade-off between completeness and conciseness is extremely complicated. But do beware that, based on only my understanding of your post, you might still be missing important insights about the distinction between numerical linear algebra and abstract linear algebra.
I honestly don’t think the tradeoff is real (but please tell me if you don’t find my reasons compelling). If I study category theory next and it does some cool stuff with the base map, I won’t reject that on the basis of it contradicting this book. Ditto if I actually use LA and want to do calculations. The philosophical understanding that matrix-vector multiplication isn’t ultimately a thing can peacefully coexist with me doing matrix-vector multiplication whenever I want to. Just like the understanding that the natural number 1 is a different object from the integer number 1 peacefully coexists with me treating them as equal in any other context.
I don’t agree that this view is theoretically limiting (if you were meaning to imply that), because it allows any calculation that was possible before. It’s even compatible with the base map.
Just to share my two cents on the matter, the distinction between abstract vectors and maps on the one hand, and columns with numbers in them (confusingly also called vectors) and matrices on the other hand, is a central headache for Linear Algebra students across the globe (and by extension also for the lecturers). If the approach this book takes works for you then that’s great to hear, but I’m wary of `hacks’ like this that only supply a partial view of the distinction. In particular matrix-vector mulitplication is something that’s used almost everywhere, if you need several translation steps to make use of this that could be a serious obstacle. Also the base map ΦB:Fn→V that limerott mentions is of central importance from a category-theoretic point of view and is essential in certain more advanced fields, for example in differential geometry. I’m therefore not too keen on leaving it out of a Linear Algebra introduction.
Unfortunately I don’t really know what to do about this, like I said this topic has always caused major confusion and the trade-off between completeness and conciseness is extremely complicated. But do beware that, based on only my understanding of your post, you might still be missing important insights about the distinction between numerical linear algebra and abstract linear algebra.
I honestly don’t think the tradeoff is real (but please tell me if you don’t find my reasons compelling). If I study category theory next and it does some cool stuff with the base map, I won’t reject that on the basis of it contradicting this book. Ditto if I actually use LA and want to do calculations. The philosophical understanding that matrix-vector multiplication isn’t ultimately a thing can peacefully coexist with me doing matrix-vector multiplication whenever I want to. Just like the understanding that the natural number 1 is a different object from the integer number 1 peacefully coexists with me treating them as equal in any other context.
I don’t agree that this view is theoretically limiting (if you were meaning to imply that), because it allows any calculation that was possible before. It’s even compatible with the base map.