Construction
Given a set, we can construct a free -module over . The module is simply the direct sum of copies of, often denoted . We give a concrete realization of this direct sum, denoted by, as follows:
- Carrier: contains the functions such that for cofinitely many (all but finitely many) .
- Addition: for two elements, we define by .
- Inverse: for, we define by .
- Scalar multiplication: for, we define by .
A basis for is given by the set where
(a variant of the Kronecker delta and a particular case of the indicator function, for the set ).
Define the mapping by . This mapping gives a bijection between and the basis vectors . We can thus identify these sets. Thus may be considered as a linearly independent basis for .
Read more about this topic: Free Module
Famous quotes containing the word construction:
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“When the leaders choose to make themselves bidders at an auction of popularity, their talents, in the construction of the state, will be of no service. They will become flatterers instead of legislators; the instruments, not the guides, of the people.”
—Edmund Burke (1729–1797)