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 (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 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.
 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

and moreover

Polars 

Suppose now that  is a weakly continuous linear operator between topological vector spaces  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 spaces 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 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. 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, 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  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 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
Linear algebra
Linear functionals