Editing Linear algebra (section)

===Dual map===
{{main|Transpose of a linear map}}

Let 
:<math>f:V\to W</math>
be a linear map. For every linear form {{mvar|h}} on {{mvar|W}}, the [[composite function]] {{math|''h'' ∘ ''f''}} is a linear form on {{mvar|V}}. This defines a linear map
:<math>f^*:W^*\to V^*</math>
between the dual spaces, which is called the '''dual''' or the '''transpose''' of {{mvar|f}}.

If {{mvar|V}} and {{mvar|W}} are finite-dimensional, and {{mvar|M}} is the matrix of {{mvar|f}} in terms of some ordered bases, then the matrix of {{mvar|f*}} over the dual bases is the [[transpose]] {{math|''M''<sup>T</sup>}} of {{mvar|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 
:<math>\langle h^\mathsf T , M \mathbf v\rangle = \langle h^\mathsf T M, \mathbf v\rangle.</math>
To highlight this symmetry, the two members of this equality are sometimes written 
:<math>\langle h^\mathsf T \mid M \mid \mathbf v\rangle.</math>