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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the transpose of a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.


Definition

Let X^ denote the algebraic dual space of a vector space X. Let X and Y be vector spaces over the same field \mathcal. If u : X \to Y is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
, then its algebraic adjoint or dual, is the map ^ u : Y^ \to X^ defined by f \mapsto f \circ u. The resulting functional ^ u(f) := f \circ u is called the pullback of f by u. The continuous dual space of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) X is denoted by X^. If X and Y are TVSs then a linear map u : X \to Y is weakly continuous if and only if ^ u\left(Y^\right) \subseteq X^, in which case we let ^t u : Y^ \to X^ denote the restriction of ^ u to Y^. The map ^t u is called the transpose or algebraic adjoint of u. The following identity characterizes the transpose of u: \left\langle ^t u(f), x \right\rangle = \left\langle f, u(x) \right\rangle \quad \text f \in Y ^ \text x \in X, where \left\langle \cdot, \cdot \right\rangle is the natural pairing defined by \left\langle z, h \right\rangle := z(h).


Properties

The assignment u \mapsto ^t u produces an injective linear map between the space of linear operators from X to Y and the space of linear operators from Y^ to X^. If X = Y then the space of linear maps is an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that ^t (u v) = ^t v ^t u. In the language of
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over \mathcal to itself. One can identify ^t \left(^t u\right) with u using the natural injection into the double dual. * If u : X \to Y and v : Y \to Z are linear maps then ^t (v \circ u) = ^t u \circ ^t v * If u : X \to Y is a ( surjective) vector space isomorphism then so is the transpose ^t u : Y^ \to X^. * If X and Y are normed spaces then \, x\, = \sup_ \left, x^(x) \ \quad \text x \in X and if the linear operator u : X \to Y is bounded then the operator norm of ^t u is equal to the norm of u; that is \, u\, = \sup \left\.


Polars

Suppose now that u : X \to Y is a weakly continuous linear operator between
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s X and Y with continuous dual spaces X^ and Y^, respectively. Let \langle \cdot, \cdot \rangle : X \times X^ \to \Complex denote the canonical dual system, defined by \left\langle x, x^ \right\rangle = x^ x where x and x^ are said to be if \left\langle x, x^ \right\rangle = x^ x = 0. For any subsets A \subseteq X and S^ \subseteq X^, let A^ = \left\ \qquad \text \qquad S^ = \left\ denote the () (resp. ). * If A \subseteq X and B \subseteq Y are convex, weakly closed sets containing the origin then ^t u\left(B^\right) \subseteq A^ implies u(A) \subseteq B. * If A \subseteq X and B \subseteq Y then (A) = \left(^t u\right)^\left(A^\right) and u(A) \subseteq B \quad \text \quad ^t u\left(B^\right) \subseteq A^. * If X and Y are locally convex then \operatorname ^t u = \left(\operatorname u\right)^.


Annihilators

Suppose X and Y are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s and u : X \to Y is a weakly continuous linear operator (so \left(^t u\right)\left(Y^\right) \subseteq X^). Given subsets M \subseteq X and N \subseteq X^, define their (with respect to the canonical dual system) by :\begin M^ :&= \left\ \\ &= \left\ \qquad \text x^(M) := \left\ \end and :\begin ^ N :&= \left\ \\ &= \left\ \qquad \text N(x) := \left\ \\ \end * The kernel of ^t u is the subspace of Y^ orthogonal to the image of u: \ker ^t u = (\operatorname u)^ * The linear map u is injective if and only if its image is a weakly dense subset of Y (that is, the image of u is dense in Y when Y is given the weak topology induced by \operatorname ^t u). * The transpose ^t u : Y^ \to X^ is continuous when both X^ and Y^ are endowed with the weak-* topology (resp. both endowed with the strong dual topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets). * ( Surjection of Fréchet spaces): If X and Y are Fréchet spaces then the continuous linear operator u : X \to Y is surjective if and only if (1) the transpose ^t u : Y^ \to X^ is injective, and (2) the image of the transpose of u is a weakly closed (i.e. weak-* closed) subset of X^.


Duals of quotient spaces

Let M be a closed vector subspace of a Hausdorff locally convex space X and denote the canonical quotient map by \pi : X \to X / M \quad \text \quad \pi(x) := x + M. Assume X / M is endowed with the quotient topology induced by the quotient map \pi : X \to X / M. Then the transpose of the quotient map is valued in M^ and ^t \pi : (X / M)^ \to M^ \subseteq X^ is a TVS-isomorphism onto M^. If X is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
then ^t \pi : (X / M)^ \to M^ is also an isometry. Using this transpose, every continuous linear functional on the quotient space X / M is canonically identified with a continuous linear functional in the annihilator M^ of M.


Duals of vector subspaces

Let M be a closed vector subspace of a Hausdorff locally convex space X. If m^ \in M^ and if x^ \in X^ is a continuous linear extension of m^ to X then the assignment m^ \mapsto x^ + M^ induces a vector space isomorphism M^ \to X^ / \left(M^\right), which is an isometry if X is a Banach space. Denote the inclusion map by \operatorname : M \to X \quad \text \quad \operatorname(m) := m \quad \text m \in M. The transpose of the inclusion map is ^t \operatorname : X^ \to M^ whose kernel is the annihilator M^ = \left\ and which is surjective by the
Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
. This map induces an isomorphism of vector spaces X^ / \left(M^\right) \to M^.


Representation as a matrix

If the linear map u is represented by the matrix A with respect to two bases of X and Y, then ^t u is represented by the transpose matrix A^T with respect to the dual bases of Y^ and X^, hence the name. Alternatively, as u is represented by A acting to the right on column vectors, ^t u is represented by the same matrix acting to the left on row vectors. These points of view are related by the canonical inner product on \R^n, which identifies the space of column vectors with the dual space of row vectors.


Relation to the Hermitian adjoint

The identity that characterizes the transpose, that is, \left ^(f), x\right= , u(x) is formally similar to the definition of the
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \l ...
, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map Y^ \to X^ and is defined for linear maps between any vector spaces X and Y, without requiring any additional structure. The Hermitian adjoint maps Y \to X and is only defined for linear maps between Hilbert spaces, as it is defined in terms of the inner product on the Hilbert space. The Hermitian adjoint therefore requires more mathematical structure than the transpose. However, the transpose is often used in contexts where the vector spaces are both equipped with a nondegenerate bilinear form such as the Euclidean
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
or another inner product. In this case, the nondegenerate bilinear form is often used implicitly to map between the vector spaces and their duals, to express the transposed map as a map Y \to X. For a complex Hilbert space, the inner product is sesquilinear and not bilinear, and these conversions change the transpose into the adjoint map. More precisely: if X and Y are Hilbert spaces and u : X \to Y is a linear map then the transpose of u and the Hermitian adjoint of u, which we will denote respectively by ^t u and u^, are related. Denote by I : X \to X^ and J : Y \to Y^ the canonical antilinear isometries of the Hilbert spaces X and Y onto their duals. Then u^ is the following composition of maps: :Y \overset Y^* \overset X^* \overset X


Applications to functional analysis

Suppose that X and Y are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s and that u : X \to Y is a linear map, then many of u's properties are reflected in ^t u. * If A \subseteq X and B \subseteq Y are weakly closed, convex sets containing the origin, then ^t u\left(B^\right) \subseteq A^ implies u(A) \subseteq B. * The null space of ^t u is the subspace of Y^ orthogonal to the range u(X) of u. * ^t u is injective if and only if the range u(X) of u is weakly closed.


See also

* * * * * *


References


Bibliography

* * * * {{Functional analysis Functional analysis Linear algebra Linear functionals