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...

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

/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...

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...

..,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...

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...

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...

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...

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...

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

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

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

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...

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

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

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

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...

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

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

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...

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

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...

In linear algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria...

condition exactly are the ones that provide the spanning and linear independence properties with linear combinations ℓ ( b ) = c 0 b 0 + c 1 b 1 + … c n − 1 b...

implies that x would be in the D-span of Y, contradicting the D-linear independence of X, therefore x • I ≠ 0. Since U is simple, we have: x • I = U...

extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of e {\displaystyle e} , to show that π {\displaystyle...

statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation...