Basis (linear algebra): Difference between revisions
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In [[linear algebra]], a '''basis''' for a [[vector space]] <math>V</math> is a set of [[vector]]s in <math>V</math> such that every vector in <math>V</math> can be written uniquely as a finite [[linear combination]] of vectors in the space. Every nonzero vector space has a basis, and | In [[linear algebra]], a '''basis''' for a [[vector space]] <math>V</math> is a set of [[vector]]s in <math>V</math> such that every vector in <math>V</math> can be written uniquely as a finite [[linear combination]] of vectors in the basis. One may think of the vectors in a basis as building blocks from which all other vectors in the space can be assembled. This is similar to viewing the [[prime number]]s as building blocks from which [[positive]] [[integer]]s can be assembled. | ||
Every nonzero vector space has a basis, and in fact, infinitely many different bases. This result is of paramount importance in the theory of vector spaces, in the same way that the [[unique factorization|unique factorization theorem]] is of fundamental importance in the study of integers. For instance, a the existence of a ''finite'' basis for a vector space provides the space with a [[invertible]] [[linear transformation]] to [[Euclidean space]], given by taking the [[coordinate (mathematics)/advanced|coordinates]] of a vector with respect to a basis. Through this transformation, every finite dimensional vector space can be considered to be essentially "the same as" a Euclidean space, just with different labels for the vectors and operations. | |||
The term '''basis''' is also used in [[abstract algebra]], specifically in the theory of [[free module]]s. For more on this use of the term, see the [[basis (mathematics)/advanced|advanced subpage]]. | |||
Revision as of 13:22, 25 November 2008
In linear algebra, a basis for a vector space is a set of vectors in such that every vector in can be written uniquely as a finite linear combination of vectors in the basis. One may think of the vectors in a basis as building blocks from which all other vectors in the space can be assembled. This is similar to viewing the prime numbers as building blocks from which positive integers can be assembled.
Every nonzero vector space has a basis, and in fact, infinitely many different bases. This result is of paramount importance in the theory of vector spaces, in the same way that the unique factorization theorem is of fundamental importance in the study of integers. For instance, a the existence of a finite basis for a vector space provides the space with a invertible linear transformation to Euclidean space, given by taking the coordinates of a vector with respect to a basis. Through this transformation, every finite dimensional vector space can be considered to be essentially "the same as" a Euclidean space, just with different labels for the vectors and operations.
The term basis is also used in abstract algebra, specifically in the theory of free modules. For more on this use of the term, see the advanced subpage.