Editing Transpose (section)

== Transposes of linear maps and bilinear forms ==
{{See also|Transpose of a linear map}}

As the main use of matrices is to represent linear maps between [[finite-dimensional vector space]]s, the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps.

This leads to a much more general definition of the transpose that works on every linear map, even when linear maps cannot be represented by matrices (such as in the case of infinite dimensional vector spaces). In the finite dimensional case, the matrix representing the transpose of a linear map is the transpose of the matrix representing the linear map, independently of the [[basis (linear algebra)|basis]] choice.

=== Transpose of a linear map ===
{{Main|Transpose of a linear map}}

Let {{math|''X''<sup>#</sup>}} denote the [[algebraic dual space]] of an {{mvar|R}}-[[Module (mathematics)|module]] {{mvar|X}}. 
Let {{mvar|X}} and {{mvar|Y}} be {{mvar|R}}-modules. 
If {{math|''u'' : ''X'' → ''Y''}} is a [[linear map]], then its '''algebraic adjoint''' or '''dual''',{{sfn | Schaefer | Wolff | 1999 | p=128}} is the map {{math|''u''<sup>#</sup> : ''Y''<sup>#</sup> &rarr; ''X''<sup>#</sup>}} defined by {{math|''f'' {{mapsto}} ''f'' ∘ ''u''}}. 
The resulting functional {{math|''u''<sup>#</sup>(''f'')}} is called the '''[[pullback (differential geometry)|pullback]]''' of {{mvar|f}} by {{mvar|u}}. 
The following [[Relation (math)|relation]] characterizes the algebraic adjoint of {{mvar|u}}<ref>{{harvnb|Halmos|1974|loc=§44}}</ref> 
:{{math|{{angbr|''u''<sup>#</sup>(''f''), ''x''}} {{=}} {{angbr|''f'', ''u''(''x'')}}}} for all {{math|''f'' ∈ ''Y''<sup>#</sup>}} and {{math|''x'' ∈ ''X''}}
where {{math|{{angbr|&bull;, &bull;}}}} is the [[natural pairing]] (i.e. defined by {{math|{{angbr|''h'', ''z''}} :{{=}} ''h''(''z'')}}). 
This definition also applies unchanged to left modules and to vector spaces.<ref>{{harvnb|Bourbaki|1989|loc=II §2.5 }}</ref>

The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint ([[#Adjoint|below]]).

The [[continuous dual space]] of a [[topological vector space]] (TVS) {{mvar|X}} is denoted by {{math|''X''{{big|{{'}}}}}}. 
If {{mvar|X}} and {{mvar|Y}} are TVSs then a linear map {{math|''u'' : ''X'' → ''Y''}} is '''weakly continuous''' if and only if {{math|''u''<sup>#</sup>(''Y''{{big|{{'}}}}) &sube; ''X''{{big|{{'}}}}}}, in which case we let {{math|<sup>t</sup>''u'' : ''Y''{{big|{{'}}}} &rarr; ''X''{{big|{{'}}}}}} denote the restriction of {{math|''u''<sup>#</sup>}} to {{math|''Y''{{big|{{'}}}}}}. 
The map {{math|<sup>t</sup>''u''}} is called the '''transpose'''{{sfn | Trèves | 2006 | p=240}} of {{mvar|u}}. 

If the matrix {{math|'''A'''}} describes a linear map with respect to [[basis (linear algebra)|bases]] of {{mvar|V}} and {{mvar|W}}, then the matrix {{math|'''A'''<sup>T</sup>}} describes the transpose of that linear map with respect to the [[Dual basis|dual bases]].

=== Transpose of a bilinear form ===
{{main|Bilinear form}}

Every linear map to the dual space {{math|''u'' : ''X'' → ''X''<sup>#</sup>}} defines a bilinear form {{math|''B'' : ''X'' × ''X'' → ''F''}}, with the relation {{math|''B''(''x'', ''y'') {{=}} ''u''(''x'')(''y'')}}. 
By defining the transpose of this bilinear form as the bilinear form {{mvar|<sup>t</sup>''B''}} defined by the transpose {{math|<sup>t</sup>''u'' : ''X''<sup>##</sup> → ''X''<sup>#</sup>}} i.e. {{math|<sup>t</sup>''B''(''y'', ''x'') {{=}} <sup>t</sup>''u''(Ψ(''y''))(''x'')}}, we find that {{math|''B''(''x'', ''y'') {{=}} <sup>t</sup>''B''(''y'', ''x'')}}. 
Here, {{mvar|Ψ}} is the natural [[homomorphism]] {{math|''X'' → ''X''<sup>##</sup>}} into the [[double dual]].

=== Adjoint ===
{{distinguish|Hermitian adjoint}}

If the vector spaces {{mvar|X}} and {{mvar|Y}} have respectively [[nondegenerate form|nondegenerate]] [[bilinear form]]s {{math|''B''<sub>''X''</sub>}} and {{math|''B''<sub>''Y''</sub>}}, a concept known as the '''adjoint''', which is closely related to the transpose, may be defined:

If {{nowrap|{{math|''u'' : ''X'' → ''Y''}}}} is a [[linear map]] between [[vector space]]s {{mvar|X}} and {{mvar|Y}}, we define {{mvar|g}} as the '''adjoint''' of {{mvar|u}} if {{nowrap|{{math|''g'' : ''Y'' → ''X''}}}} satisfies
:<math>B_X\big(x, g(y)\big) = B_Y\big(u(x), y\big)</math> for all {{math|''x'' &isin; ''X''}} and {{math|''y'' &isin; ''Y''}}.

These bilinear forms define an [[isomorphism]] between {{mvar|X}} and {{math|''X''<sup>#</sup>}}, and between {{mvar|''Y''}} and {{math|''Y''<sup>#</sup>}}, resulting in an isomorphism between the transpose and adjoint of {{mvar|u}}. 
The matrix of the adjoint of a map is the transposed matrix only if the [[basis (linear algebra)|bases]] are [[Orthonormality|orthonormal]] with respect to their bilinear forms. 
In this context, many authors however, use the term transpose to refer to the adjoint as defined here.

The adjoint allows us to consider whether {{nowrap|{{math|''g'' : ''Y'' → ''X''}}}} is equal to {{nowrap|{{math|''u''<sup> −1</sup> : ''Y'' → ''X''}}}}. 
In particular, this allows the [[orthogonal group]] over a vector space {{mvar|X}} with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps {{nowrap|{{math|''X'' → ''X''}}}} for which the adjoint equals the inverse.

Over a complex vector space, one often works with [[sesquilinear form]]s (conjugate-linear in one argument) instead of bilinear forms. 
The [[Hermitian adjoint]] of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal.