Thank you for this response. I found it very illuminating.

To clear up: yes, I was asking specifically about [working] mathematicians, as you correctly interpreted.

]]>In general, any linear transformation from a space with basis to a space with basis has a matrix representation, and the coefficients are determined by

where the decomposition of as a linear combination of the exists and is unique because the latter form a basis of . That’s all a matrix is: a list of the coefficients of this decomposition with respect to the choices of bases for and .

To your parenthetical question, bases come from all sorts of places, depending on the application. In this case, I’ve taken a particular basis as part of the setup of my example. If you start with a basis of you automatically get a “dual basis” of the dual space . For finite-dimensional spaces (which I’ve also assumed that is here), a basis is always guaranteed to exist, though for infinite-dimensional spaces the question gets a little trickier.

]]>BTW, university professors have no say in how SC runs its K-12 programs. Believe me, we’ve tried. And my blog is ComputationalThought.

]]>since the target is 1-dimensional we don’t need to count its basis

So why call it “matrix coefficients” for a linear transform? Why not just a list of coefficients that go along some basis? (but how do you get the basis?)

]]>Like any linear function, we can write down matrix coefficients .

How?

]]>The only major difference between the way Euclid seems to do geometry and they way most mathematicians now would think about it is that he conceived of his geometrical entities as describing “real” things, but now we’re a lot more careful about what “real” means.

In the fine details, Euclid tends to be more wordy than modern proofs might be, but that’s more of a stylistic difference, and nothing that a working mathematician wouldn’t be able to figure out given a little thought.

As to the snarky comment below, claiming to come from the writer of the nonexistent “professorcurmudgeon” blog, this question was asking about mathematicians rather than students. He also doesn’t mention what level of students at what sort of school he’s talking about. That said, it wouldn’t surprise me that students in general know less about the history and philosophy of mathematics than professional mathematicians do. On the other hand, the fault is far more likely to lie with their teachers who have evidently spent 35+ years bitching about the situation rather than working to fix it.

]]>You obviously have more than enough knowledge to actually do the math. But when you read the *Elements*, does it make sense to you? The actual way it is written, and are you able to follow the proofs easily?

Do mathematicians bother going through that stuff anymore, or is it a waste of time? Here, I don’t mean the fundamentals of Euclidean geometry; I mean specifically the way it is set out in the *Elements*. Or is the format just simply too out of date to be useful?

b) the logic really is reverse. If you write it like you write, it seems like there is a ‘high priest of what is allowed in nature’ who cannot be questioned.

Personally I would write roughly either of the following two:

– … the coulomb Field (which was introduced before) can be written as E=-grad phi (direct ‘proof’ with grad 1/r = – vec (r)/r^3)

So without going the round with curl =0. Or

– In Electrostatics no Energie enters the System, therefore the closed line Integral of the Electrical field (which represents the work on a testcharge) must be zero. This implies existence of phi ( either give a proof or refer to math book)

Well, probably you will find these 2 suggestions objectionable.and

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