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:
“Theres no art
To find the minds construction in the face:
He was a gentleman on whom I built
An absolute trust.”
—William Shakespeare (15641616)
“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 authors 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)