Linear Functional

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right. In general, if V is a vector space over a field k, then a linear functional ƒ is a function from V to k, which is linear:

for all
for all

The set of all linear functionals from V to k, Homk(V,k), is itself a vector space over k. This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space. It is often written V* or when the field k is understood.

Read more about Linear FunctionalContinuous Linear Functionals, Properties, Visualizing Linear Functionals, Dual Vectors and Bilinear Forms

Other articles related to "linear functional, linear, functionals":

Linear Functional - Bases in Finite Dimensions - The Dual Basis and Inner Product
... In higher dimensions, this generalizes as follows where is the Hodge star operator. ...
... In linear algebra, a one-form on a vector space is the same as a linear functional on the space ... usually distinguishes the one-forms from higher-degree multilinear functionals on the space ... For details, see linear functional ...
Bounded Variation - Formal Definition - BV Functions of Several Variables
... a finite vector Radon measure such that the following equality holds that is, defines a linear functional on the space of continuously differentiable vector functions of compact support contained in the ... two definitions are equivalent since if then therefore defines a continuous linear functional on the space ... Since as a linear subspace, this continuous linear functional can be extended continuously and linearily to the whole by the Hahn–Banach theorem i.e ...
Banach Space - Theorems and Properties
... Then, B(X, Y) = {T X → Y
Algebra Representation - Weights
... homomorphism from the algebra to its underlying ring a linear functional that is also multiplicative) ... As the pairing is bilinear, "which multiple" is an A-linear functional of A (an algebra map A → R), namely the weight ... vector is a vector such that for all elements for some linear functional – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar ...

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