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...

study of differential equations, where it can sometimes show the linear independence of a set of solutions. The Wrońskian of two differentiable functions...

/ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid...

vn are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary...

this case, the finite subset can be taken as B itself to check for linear independence in the above definition. It is often convenient or even necessary...

..,x_{n})=\sup R} is the linear independence measure. If the x 1 , . . . , x n {\displaystyle x_{1},...,x_{n}} are linearly dependent over Q {\displaystyle...

finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential...

coefficients of the linear combination. Linear independence The elements of a subset G of a F-vector space V are said to be linearly independent if no element...

{2}}} . Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of L {\displaystyle \mathbb...

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example...

real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the...

a linear combination of these transcendentals. The converse is false: not every algebraic matroid has a linear representation. Linear independence Transcendental...

Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...

functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the...

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory...

matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations...

as an inverse limit of all such maps. Dedekind's theorem on the linear independence of characters. Let M be a monoid and k an integral domain, viewed...

squares, linear least squares Gram–Schmidt process Woodbury matrix identity Vector space Linear combination Linear span Linear independence Scalar multiplication...

the linear independence of the blocks. An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation...

is the affine analog of linear independence of vectors, or more precisely of maximal rank), and d + 1 points in general linear position in affine d-space...

Subspaces are then built up from the points using the definition of linear independence of sets of points. A vector subspace of algebraic dimension d + 1...

(mathematical logic), logical independence Independence (probability theory), statistical independence Linear independence Independence (1976 film), a docudrama...

matroid is a mathematical structure that generalizes the notion of linear independence from vector spaces to arbitrary sets. If an optimization problem...

Zbl 0784.05002. Whitney, Hassler (1935), "On the abstract properties of linear independence", American Journal of Mathematics, 57 (3): 509–533, doi:10.2307/2371182...

x=2y-7} ). linear form A linear map from a vector space to its field of scalars linear independence Property of being not linearly dependent. linear map A...

∂ f ( x ∗ ) {\displaystyle \partial f(x^{*})} is exactly balanced by a linear sum of forces ∂ h j ( x ∗ ) {\displaystyle \partial h_{j}(x^{*})} and ∂...

E_{j}^{a}\ } with   E c j   {\displaystyle \ E_{c}^{j}\ } and using the linear independence of the   E a k   {\displaystyle \ E_{a}^{k}\ } ). It is then easy...

matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid...

algorithm (incomplete) Basis (linear algebra) Burrows–Abadi–Needham logic Direct proof Generating a vector space Linear independence Polynomial Proof Pumping...

and formally proves the linear space properties for these operations. ... He then develops the theory of linear independence in a way that is astonishingly...