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Transpose of a linear map
In linear algebra, the transpose of a linear map between two vector spaces is an induced map between the dual spaces of the two vector spaces. The transpose of a linear map is often used to study the original linear map. This concept is generalized by adjoint functors.
==Definition==
If is a linear map, then the ''transpose''〔Treves (1999) p. 240〕 (or ''dual'', or ''adjoint''〔Schaefer (1999) p. 128〕), denoted by or by , is defined to be
:
for every . The resulting functional ''f''∗(''φ'') in ''V''∗ is called the ''pullback'' of ''φ'' along ''f''.
The following identity, which characterizes the transpose, holds for all and :
:
where the bracket () on the left is the duality pairing of ''Vs dual space with ''V'', and that on the right is the same with ''W''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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