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) = λf(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) = a₁x₁ + a₂x₂ + ⋯ + a_nx_n in the coordinates x₁, x₂,..., _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

y₁ = a₁₁x₁ + a₁₂x₂ + ⋯ + a_1nx_n

y₂ = a₂₁x₁ + a₂₂x₂ + ⋯ + a_2nx_n

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

y_n = a_n1x₁ + a_n2x₂ + ⋯ + a_nnx_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) = (A₁, x)a₁ + (A₂, x)a₂ + ⋯ + (A_n, x)a_n

where A₁, A₂,..., A_n, and a₁, a₂,..., 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.