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If such a linear dependence exists with at least a nonzero component, then the n vectors are linearly dependent. Linear dependencies among v 1 , ..., v n form a vector space. If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous system of linear equations , with the coordinates of the ...
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.
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 matrix [] has rank 2: the first two columns are linearly independent, so the rank is at least 2, but since the third is a linear combination of the first two (the first column plus the second), the three columns are linearly dependent so the rank must be less than 3.
Other examples include independent, unstructured, M-dependent, and Toeplitz. In exploratory data analysis, the iconography of correlations consists in replacing a correlation matrix by a diagram where the "remarkable" correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).
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 .
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 ...
If is a non-empty set with a dependence relation , then always has a basis with respect to . Furthermore, any two bases of X {\displaystyle X} have the same cardinality . If a S {\displaystyle a\triangleleft S} and S ⊆ T {\displaystyle S\subseteq T} , then a T {\displaystyle a\triangleleft T} , using property 3. and 1.