Linear form-Cedar Herald Index

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).

If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , or, when the field is understood, ; other notations are also used, such as , or When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).

Linear form

Examples

The constant zero function, mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (that is, its range is all of ). Indexing into a vector: The second element of a three-vector is given by the one-form That is, the second element of is Mean: The mean element of an -vector is given by the one-form That is, Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location. Net present value of a net cash flow, is given by the one-form where is the discount rate. That is,

Linear form

Linear functionals in Rn

Suppose that vectors in the real coordinate space are represented as column vectors

Linear form

For each row vector there is a linear functional defined by

and each linear functional can be expressed in this form.

This can be interpreted as either the matrix product or the dot product of the row vector and the column vector :

Trace of a square matrix

The trace of a square matrix is the sum of all elements on its main diagonal. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a vector space from the set of all matrices. The trace is a linear functional on this space because and for all scalars and all matrices

(Definite) Integration

Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral

is a linear functional from the vector space of continuous functions on the interval to the real numbers. The linearity of follows from the standard facts about the integral:

Evaluation

Let denote the vector space of real-valued polynomial functions of degree defined on an interval If then let be the evaluation functional

The mapping is linear since

If are distinct points in then the evaluation functionals form a basis of the dual space of ( proves this last fact using Lagrange interpolation).

Non-example

A function having the equation of a line with (for example, ) is a linear functional on , since it is not linear. It is, however, affine-linear.

Visualization

In finite dimensions, a linear functional can be visualized in terms of its level sets, the sets of vectors which map to a given value. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes. This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as Gravitation by .

Applications

Application to quadrature

If are distinct points in , then the linear functionals defined above form a basis of the dual space of , the space of polynomials of degree The integration functional is also a linear functional on , and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients for which

for all This forms the foundation of the theory of numerical quadrature.

In quantum mechanics

Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra–ket notation.

Distributions

In the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions.