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The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is ...
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.
An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram .
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 Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz .
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 .
LU factorization with () ops (GKO algorithm), and thus linear system solving, approximated or unstable algorithms for linear system solving in O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} . Here n {\displaystyle n} denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to ...
Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals is an undergraduate-level mathematics textbook on the theory of matroids. It was written by Victor Bryant and Hazel Perfect , and published in 1980 by Chapman & Hall.