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Determinants can be used to characterize linearly dependent vectors: is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix are linearly dependent. [38]
Linearly independent vectors in Linearly dependent vectors in a plane in . In the theory of vector spaces , a set of vectors is said to be linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector.
In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order n – 1.It was introduced in 1812 by the Polish mathematician Józef WroĊski, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
In the 2×2 case, if the coefficient determinant is zero, then the system is incompatible if the numerator determinants are nonzero, or indeterminate if the numerator determinants are zero. For 3×3 or higher systems, the only thing one can say when the coefficient determinant equals zero is that if any of the numerator determinants are nonzero ...
When n > m the determinant and volume are zero. When n = m, this reduces to the standard theorem that the absolute value of the determinant of n n-dimensional vectors is the n-dimensional volume. The Gram determinant is also useful for computing the volume of the simplex formed by the vectors; its volume is Volume(parallelotope) / n!.
This definition has the property that the Moore determinant of a matrix formed from a suitable collection of vectors of quaternions is zero if and only if the vectors are linearly dependent. See also [ edit ]
An alternant determinant is the determinant of a ... does not imply that the functions are linearly dependent in ... , then the alternant determinant is zero ...