# Linear Function of a Vector

## Linear Function of a Vector

a function *f*(x) of a vector variable x that has the following properties: (1) *f*(x + y) = *f*(x) + *f*(y) and (2) *f*(λx) = λ*f*(x), where λ is a number. A linear function of a vector in *n*-dimensional space is completely determined by the values it takes for *n* linearly independent vectors.

A scalar-valued linear function of a vector (a linear vector function that takes numerical values) is also called a linear functional. A linear functional in *n*-dimensional space is given by a linear form *f*(x) = *a*_{1}*x*_{1} + *a*_{2}*x*_{2} + ⋯ + *a _{n}x_{n}* in the coordinates

*x*

_{1},

*x*

_{2},...,

_{xn}of the vector x. The inner product of a vector x and a constant vector a,

*f*(x) = (a, x)

is an example of a linear functional. In a space with an inner product every linear functional is of this form.

A linear vector function of a vector defines a linear or affine transformation of a space and is also called a linear operator. A linear vector function y = *f*(x) of a vector in *n*-dimensional space is expressed in terms of coordinates by the formulas

*y*_{1} = *a*_{11}*x*_{1} + *a*_{12}*x*_{2} + ⋯ + *a*_{1n}*x _{n}*

*y*_{2} = *a*_{21}*x*_{1} + *a*_{22}*x*_{2} + ⋯ + *a*_{2n}*x _{n}*

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*y _{n}* =

*a*

_{n}_{1}

*x*

_{1}+

*a*

_{n}_{2}

*x*

_{2}+ ⋯ +

*a*

_{nn}x_{n}Here the numbers *a _{ij}* where

*i, j*= 1, 2,...,

*n*, form the matrix of the linear vector function of a vector. If the sum of the linear vector functions

*f*(x) and

*g*(x) of vector x is defined as the linear vector function

*f*(x) +

*g*(x) and the product of these functions as the linear vector function

*g*{

*f*(x)}, then to the sum and product of linear vector functions of a vector there correspond the sum and product of the corresponding matrices. An example of a linear vector function of a vector is the linear vector function of the form

*f*(x) = (A_{1}, x)a_{1} + (A_{2}, x)a_{2} + ⋯ + (A_{n}, x)a_{n}

where A_{1}, A_{2},..., A_{n}, and a_{1}, a_{2},..., a_{n} are constant vectors. In an *n*-dimensional space with an inner product every linear vector function of a vector can be represented in this form.

A function of several vector variables, which is linear in each of its independent variables, is called a multilinear (bilinear, trilinear, and so forth) function (of its vector variables). The scalar and vector products of two variable vectors can serve as examples of a scalar-valued bilinear function of two vectors and a vector-valued bilinear function of two vectors, respectively. Multilinear functions of vectors lead to the concept of a tensor.