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If such a linear dependence exists with at least a nonzero component, then the n vectors are linearly dependent. Linear dependencies among v 1 , ..., v n form a vector space. If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous system of linear equations , with the coordinates of the ...
In combinatorics, a matroid / ˈ m eɪ t r ɔɪ d / is a structure that abstracts and generalizes the notion of linear independence in vector spaces.There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats.
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 Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
Given that {} spans and is linearly dependent, one strategy is to remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this plan: There are some problems with this plan:
If is a non-empty set with a dependence relation , then always has a basis with respect to . Furthermore, any two bases of X {\displaystyle X} have the same cardinality . If a S {\displaystyle a\triangleleft S} and S ⊆ T {\displaystyle S\subseteq T} , then a T {\displaystyle a\triangleleft T} , using property 3. and 1.
The alternant can be used to check the linear independence of the functions ,, …, in function space.For example, let () = (), = and choose =, = /.Then the alternant is the matrix [] and the alternant determinant is .
Many finite matroids may be represented by a matrix over a field , in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an ...