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
denote the
algebraic dual space of a vector space
Let
and
be vector spaces over the same field
If
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
defined by
The resulting functional
is called the
pullback of
by
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)
is denoted by
If
and
are TVSs then a linear map
is weakly continuous if and only if
in which case we let
denote the restriction of
to
The map
is called the transpose or algebraic adjoint of
The following identity characterizes the transpose of
:
where
is the
natural pairing defined by
Properties
The assignment
produces an
injective linear map between the space of linear operators from
to
and the space of linear operators from
to
If
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
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
to itself.
One can identify
with
using the natural injection into the double dual.
* If
and
are linear maps then
* If
is a (
surjective) vector space isomorphism then so is the transpose
* If
and
are
normed spaces then
and if the linear operator
is bounded then the
operator norm of
is equal to the norm of
; that is
Polars
Suppose now that
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
and
with continuous dual spaces
and
respectively.
Let
denote the canonical
dual system, defined by
where
and
are said to be if
For any subsets
and
let
denote the () (resp. ).
* If
and
are convex, weakly closed sets containing the origin then
implies
* If
and
then
and
* If
and
are
locally convex then
Annihilators
Suppose
and
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
is a weakly continuous linear operator (so
). Given subsets
and
define their (with respect to the canonical dual system) by
:
and
:
* The
kernel of
is the subspace of
orthogonal to the image of
:
* The linear map
is
injective if and only if its image is a weakly dense subset of
(that is, the image of
is dense in
when
is given the weak topology induced by
).
* The transpose
is continuous when both
and
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
and
are
Fréchet spaces then the continuous linear operator
is
surjective if and only if (1) the transpose
is
injective, and (2) the image of the transpose of
is a weakly closed (i.e.
weak-* closed) subset of
Duals of quotient spaces
Let
be a closed vector subspace of a Hausdorff locally convex space
and denote the canonical quotient map by
Assume
is endowed with the
quotient topology induced by the quotient map
Then the transpose of the quotient map is valued in
and
is a TVS-isomorphism onto
If
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
is also an
isometry.
Using this transpose, every continuous linear functional on the quotient space
is canonically identified with a continuous linear functional in the annihilator
of
Duals of vector subspaces
Let
be a closed vector subspace of a Hausdorff locally convex space
If
and if
is a continuous linear extension of
to
then the assignment
induces a vector space isomorphism
which is an isometry if
is a Banach space.
Denote the
inclusion map by
The transpose of the inclusion map is
whose kernel is the annihilator
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
Representation as a matrix
If the linear map
is represented by the
matrix with respect to two bases of
and
then
is represented by the
transpose matrix
with respect to the dual bases of
and
hence the name.
Alternatively, as
is represented by
acting to the right on column vectors,
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
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,
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
and is defined for linear maps between any vector spaces
and
without requiring any additional structure.
The Hermitian adjoint maps
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
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
and
are Hilbert spaces and
is a linear map then the transpose of
and the Hermitian adjoint of
which we will denote respectively by
and
are related.
Denote by
and
the canonical antilinear isometries of the Hilbert spaces
and
onto their duals.
Then
is the following composition of maps:
:
Applications to functional analysis
Suppose that
and
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
is a linear map, then many of
's properties are reflected in
* If
and
are weakly closed, convex sets containing the origin, then
implies
* The null space of
is the subspace of
orthogonal to the range
of
*
is injective if and only if the range
of
is weakly closed.
See also
*
*
*
*
*
*
References
Bibliography
*
*
*
*
{{Functional analysis
Functional analysis
Linear algebra
Linear functionals