Editing Dual space (section)

=== Transpose of a linear map ===

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{{Main|Transpose of a linear map}}
If {{math|''f'' : ''V'' → ''W''}} is a [[linear map]], then the ''[[Transpose#Transpose of a linear map|transpose]]'' (or ''dual'') {{math|''f''{{i sup|∗}} : ''W''{{i sup|∗}} → ''V''{{i sup|∗}}}} is defined by
: <math>
    f^*(\varphi) = \varphi \circ f \,
  </math>
for every ''<math>\varphi \in W^*</math>''. The resulting functional ''<math>f^* (\varphi)</math>'' in ''<math>V^*</math>'' is called the ''[[pullback (differential geometry)|pullback]]'' of ''<math>\varphi</math>'' along ''<math>f</math>''.

The following identity holds for all ''<math>\varphi \in W^*</math>'' and ''<math>v \in V</math>'':
: <math>
    [f^*(\varphi),\, v] = [\varphi,\, f(v)],
  </math>
where the bracket [·,·] on the left is the natural pairing of ''V'' with its dual space, and that on the right is the natural pairing of ''W'' with its dual.  This identity characterizes the transpose,<ref>{{Harvp|Halmos|1974}} §44</ref> and is formally similar to the definition of the [[adjoint of an operator|adjoint]].

The assignment {{math|''f'' ↦ ''f''{{i sup|∗}}}} produces an [[injective]] linear map between the space of linear operators from ''V'' to ''W'' and the space of linear operators from ''W''{{i sup|∗}} to ''V''{{i sup|∗}}; this homomorphism is an [[isomorphism]] if and only if ''W'' is finite-dimensional.
If {{math|1=''V'' = ''W''}} then the space of linear maps is actually an [[algebra over a field|algebra]] under [[composition of maps]], and the assignment is then an [[antihomomorphism]] of algebras, meaning that {{math|1=(''fg''){{sup|∗}} = ''g''{{i sup|∗}}''f''{{i sup|∗}}}}.
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 ''F'' to itself.
It is possible to identify  (''f''{{i sup|∗}}){{sup|∗}} with ''f'' using the natural injection into the double dual.

If the linear map ''f'' is represented by the [[matrix (mathematics)|matrix]] ''A'' with respect to two bases of ''V'' and ''W'', then ''f''{{i sup|∗}} is represented by the [[transpose]] matrix ''A''<sup>T</sup> with respect to the dual bases of ''W''{{i sup|∗}} and ''V''{{i sup|∗}}, hence the name.
Alternatively, as ''f'' is represented by ''A'' acting on the left on column vectors, ''f''{{i sup|∗}} is represented by the same matrix acting on the right on row vectors.
These points of view are related by the canonical inner product on '''R'''<sup>''n''</sup>, which identifies the space of column vectors with the dual space of row vectors.