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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.

Def. Linear functional. A functional T is linear if

                                    T(av₁ + bv₂) = aTv₁ + bTv₂

for all vectors v₁ and v₂ and scalars a and b.

Examples.

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.

2. Let V be the vector space of n-square matrices over F. Let T:V R be the trace mapping

                                    T(A) = where matrix A = ( )

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 π_i:Rⁿ R be the i-th projection mapping i.e. for any vector X = (a₁, a₂, ..... , a_n) ε Rⁿ, π_i = a_i, the i-th coordinate of X. This mapping is linear and π_i 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

                                      T(v) = Av

where v is an n-element coordinate vector and A is a 1xn matrix representation of T. Thus the linear functional has the form

or

               T(v) =

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.

Syn. conjugate space, adjoint space

Example. Let V be column space consisting of all n-element column vectors over R. Let

 T: V F be

                                                T(v) = [ ]v

where are real numbers and v is any element in V. The row vector [ ] 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 { } is a basis for V. Then a basis for the dual space V* of V is the set of n linear functionals V* defined by

where i = 1,n; j = 1,n, giving the following n mappings:

             mapping: v₁ 1, v₂ 0, v₃ 0, ..... , v_n 0

             mapping: v₁ 0, v₂ 1, v₃ 0, ..... , v_n 0

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

              mapping: v₁ 0, v₂ 0, v₃ 0, ..... , v_n 1

The basis { } is called the basis dual to { } or the dual basis. There are infinitely many possible bases for V and each basis has a dual basis as defined above.

See

Hom(V,W). Vector space of all mxn matrices.

Vector space Hom(V,W)

Theorem 1. Let { } be a basis for V and let { } be the basis of V* (i.e. dual basis). Then for any vector u V,

and for any linear functional V*

Thus we see that the coordinates of u are and the coordinates of σ

are .

Annihilator. Let W be a subset (not necessarily a subspace) of a vector space V. A linear functional V* is called an annihilator of W if (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.

Theorem 2. Suppose V has finite dimension and W is a subspace of V. Then

1)        dimW + dim W⁰ = dim V

and

2)        W⁰⁰ = 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

            T ^t (ω) = ω T

that maps ω ε U* into ω T ε V* is called the transpose of T.

Thus [T ^t (ω)]v = ω(Tv) for every v ε V.

In summary, if T is a linear mapping from V into U, then T ^t is a linear mapping from U* into V*:

Theorem 3. Let T:V U be linear, and let A be the matrix representation of T relative to bases {v_i} of V and {u_i} of U. Then the transpose matrix A^t is the matrix representation of T ^t:U* V* relative to the bases dual to {u_i} and {v_i}.                                                                 

References

  Lipschutz. Linear Algebra. p. 249-251

  Taylor. Introduction to Functional Analysis. p. 33