Talk:Determinant

Somebody (myself, if I'll win the laziness) should add something about the formal definition of determinant (an alternating function of the rows or columns etc. ...), of which its unicity and how to compute it are consequences. --Goochelaar

...and add to that the foundation of the definition, which is something to do with multilinear functions.
Also worth mentioning that historically, the concept of determinant came before the matrix.

That would certainly be very interesting. What is the history of the concept? --AxelBoldt

I'll see what I can dig up, but briefly: a determinant was originally a property of a system of equations. When the idea of putting co-efficients into a grid came up, the term "matrix" was coined to mean "mother of the determinant", as in womb.
The determinant function is defined in terms of vector spaces. It is the only function f: F^n x F^n .... x F^n -> F that is multilinear & alternating such that f( standard basis ) = 1.
Obviously, the above needs a major amount of fleshing out....

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I rewrite the page in a format similar to trace of a matrix. Wshun

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Text moved over from Talk:Determinant mathematics

Perhaps mention of the Scalar Triple Product, a.k.a. the Box Product, is fitting in the paragraph about the volume of the parallelopiped. If only to introduce the nomenclature.

I'm not familiar with that. Is it just the determinant of three 3-vectors? --AxelBoldt

Essentially, yes. According to Advanced Engineering Mathematics by Erwin Kreysig: "The scalar triple product or mixed triple product of three vectors

   a = [a1, a2, a3],   b = [b1, b2, b3], c = [c1, c2, c3]

is denoted by (a b c) and is defined by

(a b c) = a · (b × c)."

Since the cross product can be defined as a determinant where the first row is comprised of unit vectors, it is easy to prove that the scalar triple product is the determinant of a matrix where each row is a vector. Take its absolute value, and you get a volume. Another use of the product, besides computing volumes, is to show that three 3-d vectors are linearly independent ((a b c) ≠ 0 => a, b, c are linearly independent). From what I understand, it's a dying notation because it can be described in terms of the dot and cross products, but it still has a couple of uses.

Perhaps just include mention of it on this page, and define it on a vector calc page.

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Hmmm - talk about determinants with vector entries - that really ducks what's going on, no? Which is a 2-vector (wedge of vectors) being paired with a vector. Charles Matthews