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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) = a11 + a22 + .... + ann

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

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) = a1v1 + a2v2 + .... + anvn

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.

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

T: V → F be

T(v) = [a1, a2, .... , an] v

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

fi(vj) = 1     if i = j

fi(vj) = 0     if i ≠ j

where i = 1,n; j = 1,n. This is the Kronecker delta mapping

fi(vj) = δ(i,j)

where δ(i,j) is the Kronecker delta.

More explicitly, it is the following n mappings:

f1 mapping: v1 1, v2 0, v3 0, ..... , vn 0

f2 mapping: v1 0, v2 1, v3 0, ..... , vn 0

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

fn mapping: v1 0, v2 0, v3 0, ..... , vn 1

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.

See

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,

u = f1(u)v1 + f2(u)v2 + .... + fn(u)vn

and for any linear functional σ V*

σ = σ(v1)f1 + σ(v2)f2 + .... + σ(vn)fn

Thus we see that the coordinates of u are f1(u), f2(u), .... , fn(u) and the coordinates of σ

are σ(v1), σ(v2), .... , σ(vn) .

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.

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 {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}.

References

Lipschutz. Linear Algebra. p. 249-251

Taylor. Introduction to Functional Analysis. p. 33