Transpose - Transpose of Linear Maps

Transpose of Linear Maps

If f : V → W is a linear map between vector spaces V and W with nondegenerate bilinear forms, we define the transpose of f to be the linear map tf : W → V, determined by

Here, B_V and B_W are the bilinear forms on V and W respectively. The matrix of the transpose of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms.

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

If V and W do not have bilinear forms, then the transpose of an F-linear map f : V → W is only defined as a linear map tf : W∗ → V∗ between the dual spaces of W and V.

This means that the transpose (and even the orthogonal group) can be defined abstractly, and completely without reference to matrices (nor the components thereof). If f : V → W then for any o : W → F (that is, any o belonging to W∗), if Tf(o) is defined as o composed with f then it will map V → F (that is, Tf will map W∗ to V∗). If the vector spaces have metrics then V∗ can be uniquely mapped to V, etc., such that we can immediately consider whether or not fT : W → V is equal to f −1 : W → V.