It's not that we don't have a good understanding of linear algebra at all. It's that we don't understand how to make it simple. It's like a separate technological problem than actually building the theory itself.
I'm not the person you were originally replying to, but I have taken all the appropriate classes and still find the dual space to be mostly inappropriately motivated. There is a style of person for whom the motivation is simply "given V, we can generate V* and it's a vector space, therefore it's worth studying". But that is not, IMO, sufficient. A person the subject can't make sense of that understanding the alternative: not defining it, and discarding it, and ultimately why one approach was stolen over the others.
I think in 50 years we will look back on the way pure math was written today as a great tragedy of this age that is thankfully lost to time.
> I think in 50 years we will look back on the way pure math was written today as a great tragedy of this age that is thankfully lost to time.
That could very well be true. I mean just a 100 years ago mathematics (and most education) consisted almost exclusively of the most insane drudgery imaginable. I do sometimes wonder what the world could have been like if we didn't gate contributions in math or physics behind learning classical greek.
I do think that some of the issues come down to different learning styles. I personally like getting the definition up front- it keeps me less confused, and I can properly appreciate the examples down the line. The way Axler introduces the dual space was really charming for me, and it clicked in a way that "vectors as columns, covectors as rows" never did. But that's not everyone! It's by no means everyone in pure math, and its definitely not everyone who needs to use math. I've met people far better than me who struggled just because the resources weren't tuned towards them- there's a huge gap.
I'm not the person you were originally replying to, but I have taken all the appropriate classes and still find the dual space to be mostly inappropriately motivated. There is a style of person for whom the motivation is simply "given V, we can generate V* and it's a vector space, therefore it's worth studying". But that is not, IMO, sufficient. A person the subject can't make sense of that understanding the alternative: not defining it, and discarding it, and ultimately why one approach was stolen over the others.
I think in 50 years we will look back on the way pure math was written today as a great tragedy of this age that is thankfully lost to time.