Application of Determinants
-------------------------------------------------
Applications
[edit]Linear independence
As mentioned above, the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors of the matrix are linearly dependent. Thus, determinants can be used to characterize linearly dependent vectors. For example, given two linearly independent vectors v1, v2 in R3, a third vector v3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given n functions f1(x), ..., fn(x) (supposed to be n−1 times differentiable), the Wronskian is defined to be
It is non-zero (for some x) in a specified interval if and only if the given functions and all their derivatives up to order n−1 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence.
[edit]Orientation of a basis
Main article: Orientation (vector space)
The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation.
More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2×2 or 3 × 3 matrix, this is a rotation), while if it is negative, Aswitches the orientation of the basis.
[edit]Volume and Jacobian determinant
As
Applications
[edit]Linear independence
As mentioned above, the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors of the matrix are linearly dependent. Thus, determinants can be used to characterize linearly dependent vectors. For example, given two linearly independent vectors v1, v2 in R3, a third vector v3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given n functions f1(x), ..., fn(x) (supposed to be n−1 times differentiable), the Wronskian is defined to be
It is non-zero (for some x) in a specified interval if and only if the given functions and all their derivatives up to order n−1 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence.
[edit]Orientation of a basis
Main article: Orientation (vector space)
The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation.
More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2×2 or 3 × 3 matrix, this is a rotation), while if it is negative, Aswitches the orientation of the basis.
[edit]Volume and Jacobian determinant
As