A pullback map is a map induced in the opposite direction from a function or morphism, as in a contravariant functor. It is distinct from a categorical pullback, which is an object defined by a universal property.
For a linear transformation 
between vector spaces,
the dual map 
is a pullback map. Its associated matrix
is the transpose of the matrix
of 
.
For a smooth map 
and a differential
r-form 
on 
, the pullback 
is a differential 
-form on 
defined by
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