Linear functional. Matrix representation. Dual space, conjugate space, adjoint space. Basis for dual space. Annihilator. Transpose of a linear mapping.

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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 + .... + a_nn

where matrix A = (a_ij). 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

T(v) = a₁v₁ + a₂v₂ + .... + a_nv_n

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) = [a₁, a₂, .... , a_n] v

where a₁, a₂, .... , a_n are real numbers and v is any element in V. The row vector [a₁, a₂, .... , a_n] 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, .... , v_n} is a basis for V. Then a basis for the dual space V* of V is the set of n linear functionals f₁, f₂, .... , f_n V* defined by

f_i(v_j) = 1 if i = j

f_i(v_j) = 0 if i ≠ j

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

f_i(v_j) = 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, ..... , v_n 0

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

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

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

The basis {f₁, f₂, .... , f_n} is called the basis dual to {v₁, v₂, ..... , v_n} 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 {v₁, v₂, ..... , v_n} be a basis for V and let {f₁, f₂, .... , f_n} be the basis of V* (i.e. dual basis). Then for any vector u V,

u = f₁(u)v₁ + f₂(u)v_{2 + .... +} f_n(u)v_n

and for any linear functional σ V*

σ = σ(v₁)f₁ + σ(v₂)f_{2 + .... +}σ(v_n)f_n

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

are σ(v₁), σ(v₂), .... ,σ(v_n) .

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 {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