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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.
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The concepts of dependence and independence of systems are partially generalized in numerical linear algebra by the condition number, which (roughly) measures how close a system of equations is to being dependent (a condition number of infinity is a dependent system, and a system of orthogonal equations is maximally independent and has a ...
Let T be a complete L-theory.An L-formula φ(x,y) is said to have the independence property (with respect to x, y) if in every model M of T there is, for each n = {0,1,...,n − 1} < ω, a family of tuples b 0,...,b n−1 such that for each of the 2 n subsets X of n there is a tuple a in M for which
In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra.
It should at least be mentioned that in a module over a ring, linear independence (in the sense that zero cannot be represented as a non-trivial linear combination) is not equivalent to one vector being contained in the span of the remaining elements.--80.136.131.201 18:37, 20 January 2006 (UTC)
A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. [18]