In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a linear form (also known as a linear functional, a
one-form, or a covector) is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
from a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
to its
field of
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
(often, the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s).
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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
. This space is called the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of , or sometimes the algebraic dual space, when a
topological dual space is also considered. It is often denoted ,
[ p. 19, §3.1] 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).
Examples
* The constant
zero function, mapping every vector to zero, is trivially a linear functional.
* Indexing into a vector: The second element of a three-vector is given by the one-form
That is, the second element of