where for all , and only finitely many are non-zero. Given a set map we may therefore define the -module homomorphism , called the linear extension of , such that
The above observation has a convenient reformulation in terms of category theory. Let denote the category of -modules, and the category of sets. Consider the adjoint functors , the forgetful functor that maps an -module to its underlying set, and , the free module functor that maps a set to the free -module generated by that set. To say that is right-adjoint to is the same as saying that every set map from to , the set underlying , corresponds naturally and bijectively to an -module homomorphism from to .
Similarly, given a map , we may define the bilinear extension
which is the unique bilinear map from to whose restriction to is .
Generally, for any positive integer and a map , we may define the -linear extension
quite compactly using multi-index notation: .
The notion of linear extension is typically used as a manner-of-speaking. Thus, when a multilinear map is defined explicitly in a mathematical text, the images of the basis elements are given accompanied by the phrase “by multilinear extension” or similar.
|Date of creation||2013-03-22 15:24:06|
|Last modified on||2013-03-22 15:24:06|
|Last modified by||GrafZahl (9234)|