Re: Unique Alternating Multilinear D:R^n->R
- From: Hatto von Aquitanien <abbot@xxxxxxxxxxxxxx>
- Date: Sat, 01 Jul 2006 01:33:34 -0400
Christopher J. Henrich wrote:
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.
I looked at the sample pages on Amazon. It looks like a good treatment to
read. One thing I noticed is that he speaks in terms of covariant
*vectors*, etc. I prefer to think of vectors as invariants, with
contravariant components, and bases a covariant. The little bit I know (or
recall) of dual vector spaces tells me they can be treated as linear
(differential) forms. That certainly doesn't mean Halmos is wrong. I just
find it curious that there appears to be a lot of diversity in approach
between authors regarding these topics. I also note that, from looking at
the TOC and index, he appears not to treat eigenvalues extensively. Not
that such a treatment is essential in the context. There are plenty of
other sources for that.
Perhaps Charles W. Curtis, _Linear_
_Algebra_ , _an_ _Introductory_ _Approach_ , is a good low church book.
Looks like a very good 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.
Ironically, when I looked at Curtis's book on Amazon, they tried to sell me
Edwards's book as well.
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)?
It's anti-symmetric in the sense of the Levi-Civita symbol.
http://en.wikipedia.org/wiki/Levi-Civita_symbol
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)?
By definition, it is 0.
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?
Infer in the sense that we /know/ the value without knowing the values of
the vectors? No. Can we evaluate D(V)? Yes. If the vectors are linearly
independent, the determinant will be non-zero. The matrix of basis vectors
can, in general, be viewed as a transformation matrix. If it happens to be
orthonormal, the components represent the cosines of the angles formed
between the domain and range systems. The columns represent the components
of the basis vectors of the range system in terms of the domain system, and
the rows represent the values of the domain basis vectors in terms of the
range system. I believe the cofactor matrix can be viewed as forming what
is called the 'dual' basis (or dual space), but I'm less sure of that.
--
Nil conscire sibi
.
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