i look for a axiomatic or abstract definition for matrix determinant . there are a lot of formulas and properties for matrix determinant but none of them uniquely and qualitatively describe matrix determinant. so it a definition described some basic simple properties that uniquely determine determinant is what I need.
Determinants allow us to determine whether a system of linear equations has a unique solution or not.
The determinant of a matrix is the product of the eigenvalues of that matrix. So if the eigenvectors are the principal axes, and the eigenvalues are the relative lengths of those axes, then the determinant can be interpreted as being a volume.
all of the definition is based on their application. i need a very fundamental mathematical definition.
How about the standard abstract linear algebra Theorem-Definition: Det is the unique mapping from the space of n-tuples of n-dimensional vectors (viewing a matrix as an ordered n-tuple of n-vectors) which satisfies:
1) It is lineal in any coordinate of the tuple
2) It is alternated (changing two columns change the sign), and
3) Det of the standard basis equals 1.
Here is a coordinate-free definition of the determinant of a linear operator.
Suppose V is an N-dimensional vector space and A: V -> V a linear operator. Consider the Kth exterior power of V. There are many equivalent definitions but they all end up with a space of symbols of the form x(1) ∧ x(2) ∧ . . . ∧ x(k) where x(i) are vectors in V.
The Nth exterior power is known to be a 1-dimensional vector space and A induces the linear operator on it:
x(1) ∧ x(2) ∧ . . . ∧ x(k) --> A[x(1)] ∧ A[x(2)] ∧ . . . ∧ A[x(k)]
But since the space is one dimensional this operator is a scalar. That scalar is called the determinant of A.
I would argue that the definition of determinant as a (signed) volume is actually rather fundamental. It also has a benefit of being geometric, i.e. easy to define and understand.
i would prefer an algebraic definition rather than geometric. @Daron Anderson 's answer seems valid but its too much complicated for an elementary course !
You can think of volume as not a geometric, but rather a measure theoretic notion. In measure theory, you actually compute with volumes as primitive objects. Looking at it this way, volume looks algebraic enough to me :)
Oh of course the answer as it stands is too complex for an elementary course. But there's always the chance you could unpack the definitions to see if it can be made more appropriate. Good luck!
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