Transpose of a linear map


In linear algebra, the transpose of a linear map 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, 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 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 of.
The following identity characterizes the transpose of
where is the natural pairing.

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 under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that.
In the language of 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.
Suppose now that is a continuous linear operator between topological vector spaces and with continuous dual spaces and, respectively.
For any subset of, let denote the polar of in.

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, 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 or another real 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 spaces and that is a linear map, then many of 's properties are reflected in.
  • If and are weakly closed, convex sets containing 0, 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.