section_path: Duality/Dual map
section_name: Dual map
paragraph_text: 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