Сызықтық тәуелділік және векторлардың сызықтық комбинациясы

Vectors whose linear combinations are nonzero
linear dependence of random variables

In the theory of vector spaces, a set of vectors is said to be linearly independent 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 linearly dependent. 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.

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 x over the reals has the (infinite) subset 1={1, x, x^(2), } as a basis.

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, n linearly independent vectors are required to describe all locations in n dimensional space.

If there are more vectors than dimensions, the vectors are linearly dependent. This is illustrated in the example above of three vectors in

Let be the vector space of all differentiable functions of a real variable Then the functions and in are linearly independent.

A linear dependency or linear relation among vectors 'v'1, , 'v'n is a tuple (a1, , an) with n scalar components such that

If such a linear dependence exists with at least a nonzero component, then the n vectors are linearly dependent. Linear dependencies among 'v'1, , 'v'n form a vector space. If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous system of linear equations, with the coordinates of the vectors as coefficients. A basis of the vector space of linear dependencies can therefore be computed by Gaussian elimination.

A set of vectors is said to be affinely dependent if at least one of the vectors in the set can be defined as an affine combination of the others. Otherwise, the set is called affinely independent. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent. Conversely, every linearly independent set is affinely independent. Consider a set of vectors of size each, and consider the set of augmented vectors of size each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.