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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.
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.
Proof: Let p = (p 1, p 2) and q ... vectors v 1, ... , v k are called linearly independent if ... A basis for a subspace S is a set of linearly independent vectors ...
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.
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.
For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S , one commonly uses one of the following phrases: S spans V ; S is a spanning set of V ; V is spanned or generated by S ; S is a generator set or a generating set of V .
If {v 1, v 2,...,v n} is a linearly independent list of vectors in an inner-product space , then there ... Proof of the Gram-Schmidt theorem is constructive, ...
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements.