Re: Unique Alternating Multilinear D:R^n->R

In article <VtednTkjpdlVojjZnZ2dnUVZ_uWdnZ2d@xxxxxxxxxxxxx>, Hatto von
Aquitanien <abbot@xxxxxxxxxxxxxx> wrote:

Edwards writes: "There exists a unique (that is, one and only one)
multilinear alternating function D, from n-tuples of vectors in R^n to real
numbers, such that D(e_1,...,e_n)=1." He also indicates that it is beneath
him to explain such trivialities, and that I should consult the chapter on
determinants in my Linear Algebra textbook. Well, that is not how the
Linear Algebra textbooks (I emphasize the plural) I consulted treat the
determinant. The definition for the determinant which I find easiest to
remember is the contraction of the Levi-Civita symbol with the elements of
the matrix appearing once per dimension, and putting the free indices in
ascending order.

I haven't given a lot of thought to showing these definitions are
equivalent. It may be fairly easy. Nonetheless, Edwards's definition
suggests an approach worth investigating further. Starting with his
definition certainly makes the subsequent proofs trivial. Crammer's rule
is almost embarrassingly simple in that form. Is this a fairly common
means of defining the determinant? Is there a good and concise treatment
of this approach available?

IIRC the last time I asked about this someone pointed me to a source on
Grassmann. I am very interested in pursuing that, but I don't believe it
would be wise in the immediate future to do so.

There's High Church math, which is austere and axiomatic, and there is
Low Church math, which is oriented to problems, examples, and
algorithms. Linear algebra is so important that you have to study it
from both approaches.

I think Halmos, _Finite_ - _Dimensional_ _Vector_ _Spaces_ is a
classic of the high church. Perhaps Charles W. Curtis, _Linear_
_Algebra_ , _an_ _Introductory_ _Approach_ , is a good low church book.

Textbooks also differ in how much the authors expect us to work out for
ourselves. I gather that Edwards is on the more demanding side. Given
his characterization of "D", here are some leading questions:

1. If (e_a_1, ..., e_a_n) is a permutation of (e_1 , ..., e_n), what is
D(e_a_1, ..., e_a_n)?

2. If (e_a_1, ..., e_a_n) is /not/ a permutation of (e_1 , ..., e_n),
but instead has some repeated basis elements, what is D(e_a_1, ...,
e_a_n)?

3. Given the answers to questions 1 and 2, can we infer the value of
D(v_1, ..., v_n) where the vectors v_i are arbitrary linear
combinations of the basis vectors?

--
Chris Henrich
http://www.mathinteract.com
The total lack of evidence is the surest sign that the conspiracy is working.
.

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