In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension.
A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.
Definition
A sequence of vectors from a vector space is said to be linearly dependent, if there exist scalars not all zero, such that
where denotes the zero vector.
This implies that at least one of the scalars is nonzero, say , and the above equation is able to be written as
if and if
Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.
A sequence of vectors is said to be linearly independent if it is not linearly dependent, that is, if the equation
can only be satisfied by for This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of as a linear combination of its vectors is the trivial representation in which all the scalars are zero. Even more concisely, a sequence of vectors is linearly independent if and only if can be represented as a linear combination of its vectors in a unique way.
If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.
A sequence of vectors is linearly independent if and only if it does not contain the same vector twice and the set of its vectors is linearly independent.
Infinite case
An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
An indexed family of vectors is linearly independent if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said to be linearly dependent.
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 over the reals has the (infinite) subset as a basis.
Geometric examples
and are independent and define the plane P. , and are dependent because all three are contained in the same plane. and are dependent because they are parallel to each other. , and are independent because and are independent of each other and is not a linear combination of them or, equivalently, because they do not belong to a common plane. The three vectors define a three-dimensional space. The vectors (null vector, whose components are equal to zero) and are dependent since
Geographic location
A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 5 miles northeast of here." This last statement is true, but it is not necessary to find the location.
In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary to define a specific location on a plane.
Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, linearly independent vectors are required to describe all locations in -dimensional space.