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An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
One of the key motivating examples in the formulation of matroids was the notion of linear independence of vectors in a vector space: if is a finite set or multiset of vectors, and is the family of linearly independent subsets of , then (,) is a matroid.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
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 particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular. When n > m the determinant and volume are zero.
Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable. A matrix that is not diagonalizable is said to be defective.
Any other pair of linearly independent vectors of R 2, such as (1, 1) and (−1, 2), forms also a basis of R 2. More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication.
Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all orthogonal to the vector n = [6, −1, 4, −4, 0] ( n is an element of the kernel of J ), so it can be deduced that the row space consists of all vectors in R 5 {\displaystyle \mathbb {R} ^{5}} that are ...