Linear functional. Matrix representation. Dual space, conjugate space, adjoint space. Basis for dual space. Annihilator. Transpose of a linear mapping. Def. Functional. Let V be an abstract vector space over a field F. A functional T is a function T:V → F that assigns a number from field F to each vector x ε V. 1. Let V be the vector space of polynomials in t over R, the field of reals. Let T:V → R be the integral operator defined by This integral effects a linear mapping from the space of polynomials to the field of reals and hence T is a linear functional. where matrix A = (aij). That is, T assigns to a matrix A the sum of its diagonal elements. This mapping can be shown to be linear and hence T is a linear functional. 3. Let πi:Rn → R be the i-th projection mapping i.e. for any vector X = (a1, a2, ….. , an) ε Rn, πi = ai, the i-th coordinate of X. This mapping is linear and πi is a linear functional on Rn. The domain V of a linear functional T: V → F can be either infinite dimensional or finite dimensional. We will consider here only linear functionals in which the domain V is finite dimensional. Matrix representation of a linear functional whose domain is finite dimensional. Any linear mapping from one finite dimensional abstract vector space to another is represented by a matrix. A linear mapping from an n-dimensional vector space over a field F to an m-dimensional vector space over F is represented by an mxn matrix.over F. A linear functional T: V → F whose domain V is finite dimensional is a linear mapping from an n-dimensional vector space to a 1-dimensional vector space and is represented by a 1xn matrix i.e. an n-element row vector. The matrix representation of the mapping is where v is an n-element coordinate vector and A is a 1xn matrix representation of T. Thus the linear functional has the form Dual Space. If V is some abstract vector space over a field F, then the dual space of V is the vector space V* consisting of all linear functionals with domain V and range contained in F. The dual space V*, of a space V, is the vector space Hom (V,F). Linear functionals whose domain is finite dimensional and of dimension n are represented by 1xn matrices and dual space [ Hom (V,F) ] corresponds to the set of all 1xn matrices over F. If V is of dimension n then the dual space has dimension n. where a1, a2, …. , an are real numbers and v is any element in V. The row vector [a1, a2, …. , an] can be viewed as a linear operator operating on vectors in V. It is a linear functional which maps elements of V into field R. The dual space V* of V is then the vector space of all n-element row vectors. Thus row space is the dual space of column space V. Basis for Dual Space. Suppose V is some abstract vector space of dimension n over a field F. Suppose {v1, v2, …. , vn} is a basis for V. Then a basis for the dual space V* of V is the set of n linear functionals f1, f2, …. , fn V* defined by The basis {f1, f2, …. , fn} is called the basis dual to {v1, v2, ….. , vn} or the dual basis. There are infinitely many possible bases for V and each basis has a dual basis as defined above. Theorem 1. Let {v1, v2, ….. , vn} be a basis for V and let {f1, f2, …. , fn} be the basis of V* (i.e. dual basis). Then for any vector u V, Annihilator. Let W be a subset (not necessarily a subspace) of a vector space V. A linear functional f V* is called an annihilator of W if f(w) = 0 for every w W. The set of all such mappings, denoted by W0 and called the annihilator of W, is a subspace of V*. Example. Pass a line through the origin of an x-y-z Cartesian coordinate system and label it L. Let W be the set of all vectors in line L. Pass a plane through the origin of the coordinate system perpendicular to line L and label it K. Let S represent the set of all vectors in plane K. Then S is an annihilator of W. Why? If s is any vector in S and w is any vector in W then the dot product s∙w = 0. The vector s can be viewed as a linear operator (linear functional) mapping the vectors of W into the field of reals and it maps all the elements of W into zero. By the same logic W is an annihilator of S. So the annihilator W0 of W is the set S and the annihilator S0 of the set S is the set W. Transpose of a linear mapping. Let T:V → U be an arbitrary linear mapping from a vector space V into a vector space U. Now for any linear functional ω ε U*, the composition mapping ω T is a linear mapping from V into F. See Fig. 1. Thus ω T ε V*. We thus have a one-to-one correspondence between ω ε U* and ω T ε V*. The linear mapping Theorem 3. Let T:V → U be linear, and let A be the matrix representation of T relative to bases {vi} of V and {ui} of U. Then the transpose matrix At is the matrix representation of T t:U* → V* relative to the bases dual to {ui} and {vi}. Source.