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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 WroĊski, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.
The Gram–Schmidt process takes a finite, linearly independent set of vectors = {, …,} for k ≤ n and generates an orthogonal set ′ = {, …,} that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [ 1 ]
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Linear independence; From an adjective: This is a redirect from an adjective, which is a word or phrase that describes a noun, to a related word or topic.
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 three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as: