quasi
Posts:
12,018
Registered:
7/15/05


Re: Maths pedagaogy
Posted:
Mar 18, 2013 1:41 AM


Kaba wrote: >quasi wrote: >> >> In your indictment of Lang's Algebra, let's see some examples. >> Show us statements from that text for which, assuming the >> student >> >> * has the assumed prerequisites for the text >> >> * has successfully understood (and done exercises to prove >> it) for the material up to that point in the text >> >> have proofs left to the reader, but which, in your opinion, >> would actually not be trivial for the average reader, as >> qualified above. > >Here is an example from Lang's Algebra, page 572, Proposition 1.1. >This is a chapter on bilinear forms. Notation is as follows. The >__ is for the orthogonal sum, that is, a direct sum of vector >spaces such that the vector spaces are orthogonal to each other. >I will denote the bilinear form by <u, v>. The radical of the >bilinear form in vector space U is denoted by rad(U). We are only >interested on bilinear forms that are either symmetric, alternating, >or hermitian; in these cases the leftand rightradicals coincide. > >Lang's proof >
You mean Lang's stated theorem.
Theorem 1.1:
>Let E be a vector space over the field k, and let g be a form >of one of the three above types [symmetric, alternating, or >hermitian]. Suppose that E is expressed as an orthogonal sum, > > E = E_1 __ ... __ E_m. > >Then g is nondegenerate on E if and only if it is >nondegenerate on each E_i.
Below is Lang's proof (of Theorem 1.1).
>Proof: > >Elements v, w of E can be written uniquely > > v = sum_{i = 1}^m v_i, > w = sum_{i = 1}^m w_i, > >with v_i, w_i in E_i. Then > > <v, w> = sum_{i = 1}^m <v_i, m_i>, > >and <v, w> = 0 if <v_i, w_i> = 0 for each i = 1, ..., m. > >From this our assertion is obvious. QED.
I'll try to clarify ...
By definition, g is nondegenerate on E if ker(g) = 0.
Equivalently, g is degenerate on E if there exists nonzero v in E such that <v,w> = 0 for all w in E.
Suppose g is degenerate on E.
Let v in E be nonzero such that <v,w> = 0 for all w in E.
Since v is nonzero, v has some nonzero component, v_i say.
Then <v,w_i> = 0 for all w_i in E_i.
But
<v,w_i> = sum (over all j) <v_j,w_i>
= <v_i,w_i>,
hence
<v_i,w_i> = 0 for all w_i in E_i
It follows that g is degenerate on E_i.
Thus, we have the implication
g degenerate on E => g degenerate on E_i for some i.
Conversely, suppose g is degenerate on E_i for some i.
Then there exists nonzero v_i in E_i such that
<v_i,w_i> = 0 for all w_i in E_i
Letting v = v_i, we get, for all w in E,
<v,w>
= <v_i,w> = sum (over all j) <v_i,w_j>
= <v_i,w_i>
= 0
so g is degenerate on E.
Thus, we have the implication
g degenerate on E_i for some i => g degenerate on E.
It follows that g is nondegenerate on E iff g is nondegenerate 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.
>Comments > > >Whoever reads this post can measure the time on how long it >takes to understand Lang's underlying logic here. I myself am >unable to decode the last two rows of the proof, which should >be the core of the proof. > >In addition to problems with clarity, Lang does not actually >use g anywhere; he uses a dot instead. Such problems show up >when you don't polish, i.e. read through, your writing >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).
>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 >
Your theorem:
>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 >nondegenerate if and only if U and W are nondegenerate.
You're mangling Lang's terminology.
In the context of the given chapter, Lang uses the terms "degenerate" and "nondegenerate" 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.
Your proof:
> > ... <snipped> >
quasi

