structural:reformatting, linguistic:paraphrasing, information:extraction
Dual map

Let

be a linear map. For every linear form h on W, the composite function h ∘ f is a linear form on V. This defines a linear map

between the dual spaces, which is called the dual or the transpose of f.

If V and W are finite dimensional, and M is the matrix of f in terms of some ordered bases, then the matrix of f* over the dual bases is the transpose M of M, obtained by exchanging rows and columns.

If elements of vector spaces and their duals are represented by column vectors, this duality may be expressed in bra–ket notation by

For highlighting this symmetry, the two members of this equality are sometimes written