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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.
An alternant determinant is the determinant of a square alternant matrix. Generally, if f 1 , f 2 , … , f n {\displaystyle f_{1},f_{2},\dots ,f_{n}} are functions from a set X {\displaystyle X} to a field F {\displaystyle F} , and α 1 , α 2 , … , α m ∈ X {\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X} , then the ...
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.
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 ]
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 ...
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.
no LU factorization if the first (n−1) columns are linearly independent and at least one leading principal minor is zero. In Case 3, one can approximate an LU factorization by changing a diagonal entry a j j {\displaystyle a_{jj}} to a j j ± ε {\displaystyle a_{jj}\pm \varepsilon } to avoid a zero leading principal minor.