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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.
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 ...
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.
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
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A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in finite projective geometry. Let P G ( N , q ) {\displaystyle PG(N,q)} be the finite projective space of (geometric) dimension N {\displaystyle N} over the finite field F q {\displaystyle \mathbb {F} _{q}} .