Linearly Dependent & Independent

                                                  Linearly Dependent & Independent

Linearly Dependent:

A subset S of a vector space V is called linearly dependent if there exist a finite number of distinct vectors v₁, v₂, ..., v_n in S and scalars a₁, a₂, ..., a_n, not all zero, such that

Note that the zero on the right is the zero vector, not the number zero.

For any vectors u₁, u₂, ..., u_n we have that

This is called the trivial representation of 0 as a linear combination of u₁, u₂, ..., u_n, this motivates a very simple definition of both linear independence and linear dependence, for a set to be linearly dependent, there must exist a non-trivial representation of 0 as a linear combination of vectors in the set.

Linearly Independent:

A subset S of a vector space V is then said to be linearly independent if it is not linearly dependent, in other words, a set is linearly independent if the only representations of 0 as a linear combination of its vectors are trivial representations.

Note that in both definitions we also say that the vectors in the subset S are linearly dependent or linearly independent.

More generally, let V be a vector space over a field K, and let {v_i | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a family {a_j | j∈J} of elements of K, not all zero, such that

where the index set J is a nonempty, finite subset of I.

A set X of elements of V is linearly independent if the corresponding family {x}_x∈X is linearly independent.