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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 ...
An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Hollow matrix: A square matrix whose main diagonal comprises only zero elements. Integer matrix: A matrix whose entries are all integers. Logical matrix: A matrix with all entries either 0 or 1.
A linear map from a vector space to its field of scalars [8] linear independence Property of being not linearly dependent. [9] linear map A function between vector space s which respects addition and scalar multiplication. linear transformation A linear map whose domain and codomain are equal; it is generally supposed to be invertible.
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 .
In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
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.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
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.