A set of vectors is linearly dependent if one can be written as a linear combination of the others, and linearly independent otherwise. In the latter case, the vectors in the set form a basis of their span.
corresponds to a homomorphism
Although is (by hypothesis) a monomorphism in , need not be a monomorphism in .
The subset is linearly independent if is a monomorphism; otherwise, is linearly dependent.
Conversely, if we start with an (abstract!) set and a monomorphism from to , then the corresponding function from to must be a monomorphism (because the underlying-set functor is faithful). Thus, our considering only subsets of loses no generality.
Given a linear combination
this may or may not equal the zero vector? . Of course, if every is the zero scalar , then the sum must be .
The subset is linearly independent if, conversely, for every finite subset , we have for all whenever
otherwise, is linearly dependent.
Observe that the empty set is linearly independent by a vacuous implication.
In constructive mathematics, the definitions above of linear independence are all right (and still equivalent), but the definition of linear dependence as simply the negation of linear independence is unsatisfying. Furthermore, we sometimes want something stronger than mere linear independence.
If is a Heyting field, then the field structure defines a tight apartness relation . Even if is not a field, we may still suppose that it is equipped with a tight apartness, or at least some inequality relation . If we restrict attention to modules with a compatible inequality relation and homomorphisms that preserve this, then we also get an inequality relation on the hom-sets of the category . This allows us to define stronger notions of both linear dependence (which we take to be the default notion) and linear independence (to which we give a new name).
The subset is linearly dependent if is non-monic in the strong sense that there exist generalised elements such that
are equal but . Concretely, is linearly dependent if some linear combination
but at least one .
The subset is linearly free if is a regular monomorphism, or equivalently if it is monic in the strong sense that whenever . Concretely, is linearly free if
whenever at least one .
Then we have the following implications (assuming that is tight, so that holds iff fails) but not (in general) their unstated converses:
It may also be instructive to look at the logical structure of each condition: * : ; * : ; * : .