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.
is called the transpose[2] or algebraic adjoint of
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
using the natural injection into the double dual.
is a weakly continuous linear operator between topological vector spaces
denote the canonical dual system, defined by
is a weakly continuous linear operator (so
define their annihilators (with respect to the canonical dual system) by[6] and
be a closed vector subspace of a Hausdorff locally convex space
and denote the canonical quotient map by
Then the transpose of the quotient map is valued in
[6] Using this transpose, every continuous linear functional on the quotient space
is canonically identified with a continuous linear functional in the annihilator
be a closed vector subspace of a Hausdorff locally convex space
induces a vector space isomorphism
This map induces an isomorphism of vector spaces
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.
is formally similar to the definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map.
and is defined for linear maps between any vector spaces
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.
the canonical antilinear isometries of the Hilbert spaces