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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 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.
When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...
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 example of a cyclical rhythm would be day fading into night, and night brightening into day; a linear rhythm might be the flow of information from a television set. Additionally, rhythms may be nested within each other; for example, the broadcast of the local news at set intervals throughout the day, throughout the week, is an example of a ...
For modules, linear independence and spanning sets are defined exactly as for vector spaces, although "generating set" is more commonly used than that of "spanning set". Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not ...
The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V if and only if they have the same coefficients. Also ...
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors , …, in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product = , . [1]