Basis (linear algebra)

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 basis vectors.

Equivalently, a set $B$ is a basis if its elements are linearly independent and every element of $V$ is a linear combination of elements of $B$ .^[1] In other words, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

Basis vectors find applications in the study of crystal structures and frames of reference.

^ Halmos, Paul Richard (1987). Finite-Dimensional Vector Spaces (4th ed.). New York: Springer. p. 10. ISBN 978-0-387-90093-3.

[1] Halmos, Paul Richard (1987). Finite-Dimensional Vector Spaces (4th ed.). New York: Springer. p. 10. ISBN 978-0-387-90093-3.

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