Definition
A free module is a module with a basis: a linearly independent generating set.
For an -module, the set is a basis for if:
- is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ;
- is linearly independent, that is, if for distinct elements of, then (where is the zero element of and is the zero element of ).
If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module, and is said to be free of rank n, or simply free of finite rank if the cardinality is finite.
Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each .
The definition of an infinite free basis is similar, except that will have infinitely many elements. However the sum must still be finite, and thus for any particular only finitely many of the elements of are involved.
In the case of an infinite basis, the rank of is the cardinality of .
Read more about this topic: Free Module
Famous quotes containing the word definition:
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (18391894)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)