Search results
Results From The WOW.Com Content Network
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 a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is ...
A basis of linearly independent lattice vectors b 1, b 2, ..., b n can be defined by g(b j) = λ j.. The lower bound is proved by considering the convex polytope 2n with vertices at ±b j / λ j, which has an interior enclosed by K and a volume which is 2 n /n!λ 1 λ 2...λ n times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λ j along each basis ...
In general, let be a value that is to be determined numerically, in the case of this article, for example, the value of the solution function of an initial value problem at a given point. A numerical method, for example a one-step method, calculates an approximate value v ~ ( h ) {\displaystyle {\tilde {v}}(h)} for this, which depends on the ...
Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, [1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
The Gram–Schmidt process takes a finite, linearly independent set of vectors = {, …,} for k ≤ n and generates an orthogonal set ′ = {, …,} that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [ 1 ]
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
More generally, if φ satisfies a polynomial equation P(φ) = 0 where P factors into distinct linear factors over F, then it will be diagonalizable: its minimal polynomial is a divisor of P and therefore also factors into distinct linear factors. In particular one has: P = X k − 1: finite order endomorphisms of complex vector spaces are ...