Re: linear algebra - basic determinant properties, how intuitive are they?
- From: Ken Pledger <Ken.Pledger@xxxxxxxxxxxxx>
- Date: Fri, 08 Apr 2005 08:26:34 +1200
In article <1112846012.062175.267150@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"porky_pig_jr@xxxxxxxxxxx" <porky_pig_jr@xxxxxxxxxxx> wrote:
> In the intro course in Linear Algrebra,
Your word "the" is a bit strong. There are legions of linear
algebra texts, handling determinants in various ways.
> the concept of determinant is
> presented by starting with three elementary requirements or properties.
> Subsequently everything else (including the permutation-based formula
> and cofactors) is derived from those three properties which are
>
> (i) determinant of identity = 1
> (ii) determinant is linear in a single row
> (iii) exchanging two rows reverses the sign of determinant
> ....
> So I am a bit puzzled - where does this 3rd requiremeent arise from?....
I prefer the approach in the little book by C. C. MacDuffee, "The
Theory of Matrices" (reprinted by Chelsea), which gives a fine summary
of linear algebra in the traditional matrix-and-determinant form as it
was in 1933.
MacDuffee pp. 6-7 introduces determinants by a method due to K.
Hensel (1928). In his slightly old-fashioned language he says:
"It is desirable to have associated with every matrix A of [his algebra
of n x n matrices] a number theta(A) of [his field of scalars] which
would serve the purpose of an absolute value. This end would be
attained by finding a scalar function theta(A) of the elements a_rs of a
general matrix A such that
1. For every A, theta(A) is a non-constant rational integral
function [i.e. polynomial] of the a_rs of lowest degree such that
2. theta(AB) = theta(A) theta(B)."
Beginning from there he deduces the three properties which you
mentioned above, attributing them to Weierstrass; then derives from them
the definition in terms of even and odd permutations.
Ken Pledger.
.
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