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| 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.<br> | 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.<br> | ||
| Obviously, the above needs a major amount of fleshing out.... | Obviously, the above needs a major amount of fleshing out.... | ||
| ---- | |||
| I rewrite the page in a format similar to ]. I will add back the following paragraph later: ] | |||
| === Interpretation of the determinant for real vectors === | |||
| If viewed as a map that takes ''n'' vectors from '''R'''<sup>''n''</sup> and produces a real result, the determinant is ''multilinear'', i.e. linear in each of its ''n'' entries. Furthermore, it is ''anti-symmetric'', i.e. exchanging two arguments multiplies the determinant by -1. | |||
| The sign of the determinant of real vectors has a special significance because it serves to define the notion of ''orientation'' of coordinate systems. If three vectors in '''R'''<sup>3</sup> are given, then they may be oriented similarly to the three vectors (1,0,0), (0,1,0), (0,0,1), i.e. similarly to the first three fingers of the right hand, in which case their determinant will be positive, or they may be oriented similarly to (1,0,0), (0,1,0), (0,0,-1), i.e. similarly to the first three fingers of the left hand, in which case their determinant will be negative. A similar statement holds true for higher dimensions. | |||
| The ] of the determinant of real vectors is important in volume computations because it is equal to the volume of the ] spanned by those vectors. This also means that the determinant of ''n'' vectors is zero if and only if the ''n'' vectors are ]. As a consequence, if the ] ''f'' : '''R'''<sup>''n''</sup> <tt>-></tt> '''R'''<sup>''n''</sup> is represented by the matrix ''A'', and ''S'' is any ] subset of '''R'''<sup>''n''</sup>, then the volume of ''f''(''S'') is given by |det(''A'')| × volume(''S''). More generally, if the linear map ''f'' : '''R'''<sup>''n''</sup> <tt>-></tt> '''R'''<sup>''m''</sup> is represented by the ''m''-by-''n'' matrix ''A'', and ''S'' is any measurable subset of '''R'''<sup>''n''</sup>, then the ''n''-dimensional volume of ''f''(''S'') is given by √(det(''A''<sup>tr</sup> * ''A'')) × volume(''S''), where ''A''<sup>tr</sup> denotes the transpose of ''A''. | |||
Revision as of 23:00, 20 March 2003
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....
I rewrite the page in a format similar to trace of a matrix. I will add back the following paragraph later: Wshun
Interpretation of the determinant for real vectors
If viewed as a map that takes n vectors from R and produces a real result, the determinant is multilinear, i.e. linear in each of its n entries. Furthermore, it is anti-symmetric, i.e. exchanging two arguments multiplies the determinant by -1.
The sign of the determinant of real vectors has a special significance because it serves to define the notion of orientation of coordinate systems. If three vectors in R are given, then they may be oriented similarly to the three vectors (1,0,0), (0,1,0), (0,0,1), i.e. similarly to the first three fingers of the right hand, in which case their determinant will be positive, or they may be oriented similarly to (1,0,0), (0,1,0), (0,0,-1), i.e. similarly to the first three fingers of the left hand, in which case their determinant will be negative. A similar statement holds true for higher dimensions.
The absolute value of the determinant of real vectors is important in volume computations because it is equal to the volume of the parallelepiped spanned by those vectors. This also means that the determinant of n vectors is zero if and only if the n vectors are linearly dependent. As a consequence, if the linear map f : R -> R is represented by the matrix A, and S is any measurable subset of R, then the volume of f(S) is given by |det(A)| × volume(S). More generally, if the linear map f : R -> R is represented by the m-by-n matrix A, and S is any measurable subset of R, then the n-dimensional volume of f(S) is given by √(det(A * A)) × volume(S), where A denotes the transpose of A.