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Topic: Maths pedagaogy
Replies: 57   Last Post: Mar 21, 2013 9:47 PM

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 Kaba Posts: 289 Registered: 5/23/11
Re: Maths pedagaogy
Posted: Mar 18, 2013 3:00 PM

18.3.2013 8:21, quasi wrote:
>>From this our assertion is obvious. QED.
>
> I'll try to clarify ...
>
> By definition, g is non-degenerate on E if ker(g) = 0.

--8x-- (full proof)

> It follows that g is non-degenerate on E iff g is non-degenerate
> on each E_i.
>
> I agree that the proof could use more detail, but I also think
> that the student, at the level expected for learning the material
> in that textbook, should be able to supply those details.

I think our proofs illustrate that Lang abuses the word obvious:) Both
proofs are tremendous improvements over Lang's. I would summarize this
as: either provide the proof, or make it an explicit exercise; but don't
do both.

It is easy to become blind to such issues after you have already
mastered the subject.

>> In addition to problems with clarity, Lang does not actually
>> use g anywhere; he uses a dot instead. Such problems show up
>> carefully.

>
> Such problems show up when the student jumps into a chapter
> without reading the author's stated notational conventions.
>
> At the beginning of the chapter, on the bottom of page 571,
> Lang clearly states that for notational convenience, if there
> is no potential ambiguity, he will use <v,w> or even v.w as a
> shorthand for g(v,w).

That's true. However, this is still bad style-wise (which is the main
point of my replies). Such definitions should be local. For example, you
don't want to hunt for the domain of function f from 8 pages back (been
there).

>> Since I had hard time with Lang's proof, I ended making up my
>> own proof; that proof follows next. Perhaps these can be
>> compared for clarity.
>>
>> My proof
>> --------

>
>

>> Let V be a bilinear space, where the bilinear form is either
>> symmetric, alternating, or hermitian. Let U, W subset V be
>> subspaces of V, such that V = U _|_ W. Then V is
>> non-degenerate if and only if U and W are non-degenerate.

>
> You're mangling Lang's terminology.
>
> In the context of the given chapter, Lang uses the terms
> "degenerate" and "non-degenerate" as properties of a given
> bilinear form, not as properties of a space.
>
> Some other authors apply those terms in the same way as you,
> but Lang does not.

Page 573: "Instead of saying that a form is non-degenerate on E, we
shall sometimes say, by abuse of language, that E is non-degenerate".

Anyway, even if that didn't read there, since it's my theorem and proof,
I get to choose:) In practice, once you start working with bilinear
spaces, you'll quickly find the subspace-based terms more convenient.

--
http://kaba.hilvi.org

Date Subject Author
3/17/13 Paul
3/17/13 David C. Ullrich
3/17/13 Paul
3/17/13 David C. Ullrich
3/17/13 Paul
3/18/13 David C. Ullrich
3/18/13 Paul
3/18/13 David C. Ullrich
3/18/13 William Elliot
3/18/13 Paul
3/18/13 Frederick Williams
3/18/13 Paul
3/18/13 Frederick Williams
3/18/13 Frederick Williams
3/18/13 Paul
3/18/13 Frederick Williams
3/19/13 David Bernier
3/18/13 Frederick Williams
3/18/13 William Elliot
3/17/13 Kaba
3/17/13 Frederick Williams
3/17/13 David C. Ullrich
3/18/13 Kaba
3/17/13 quasi
3/17/13 Kaba
3/18/13 quasi
3/18/13 Kaba
3/19/13 quasi
3/19/13 Frederick Williams
3/19/13 Paul
3/19/13 David C. Ullrich
3/19/13 Frederick Williams
3/19/13 fom
3/20/13 David C. Ullrich
3/20/13 Paul
3/20/13 fom
3/19/13 Paul
3/20/13 Herman Rubin
3/20/13 Brian Q. Hutchings
3/21/13 Herman Rubin
3/20/13 Paul
3/21/13 Herman Rubin
3/21/13 fom
3/19/13 quasi
3/19/13 Frederick Williams
3/20/13 Paul
3/20/13 Herman Rubin
3/18/13 Shmuel (Seymour J.) Metz
3/20/13 Brian Q. Hutchings
3/17/13 fom
3/17/13 Shmuel (Seymour J.) Metz
3/18/13 Frederick Williams
3/21/13 Jesse F. Hughes
3/21/13 fom
3/21/13 Kaba
3/21/13 fom
3/21/13 fom
3/18/13 grei