Linear Independence - Definition

Definition

A finite subset of n vectors, v₁, v₂, ..., v_n, from the vector space V, is linearly dependent if and only if there exists a set of n 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.

If such scalars do not exist, then the vectors are said to be linearly independent.

Alternatively, linear independence can be directly defined as follows: a set of vectors is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions, i.e., whenever a₁, a₂, ..., a_n are scalars such that

if and only if a_i = 0 for i = 1, 2, ..., n.

A set of vectors is then said to be linearly dependent if it is not 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.

Equivalently, a family is dependent if a member is in the linear span of the rest of the family, i.e., a member is a linear combination of the rest of the family.

A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. For example, the vector space of all polynomials in x over the reals has for a basis the (infinite) subset {1, x, x2, ...}.