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Linear algebra is the branch of mathematics concerning linear equations such as: ... many problems may be interpreted in terms of linear systems. For example, let
Dimension theorem for vector spaces (vector spaces, linear algebra) Dini's theorem ; Dirac's theorems (graph theory) Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Disintegration theorem (measure theory)
In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called ...
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. The result is named after the German mathematician Ernst Steinitz.
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .
The Cayley–Hamilton theorem is an effective tool for computing the minimal polynomial of algebraic integers. For example, given a finite extension [, …,] of and an algebraic integer [, …,] which is a non-zero linear combination of the we can compute the minimal polynomial of by finding a matrix representing the -linear transformation ...
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