Basis Vectors and Basis Functions

- Overview

In mathematics, basis vectors and basis functions are both sets of elements that can be used to represent other elements through linear combinations:

• Basis vectors: A set of linearly independent vectors that can represent any vector in a vector space. The coefficients of the linear combination are called the vector's components or coordinates with respect to the basis. Basis vectors are fundamental for describing and analyzing vectors and vector spaces.
• Basis functions: A set of functions that can represent other functions in a function space. For example, basis functions can be used to describe the state of a particle in space. In numerical analysis and approximation theory, basis functions are also known as blending functions because they are used in interpolation.

A vector space can have multiple bases, but all bases have the same number of elements, called the dimension of the vector space. In function space, a complete basis set may require an infinite number of basis functions, while a vector space only needs finite vectors. 

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

Please refer to the following for more information:

• Wikipedia: Basis Function