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In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent. For instance, in the three-dimensional real vector space R3 we have the following example. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, an index set is another name for a function domain. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
 Here the first three vectors are linearly independent; but the fourth vector equals 9 times the first plus 5 times the second plus 4 times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; here we could just as well write the first vector as a linear combination of the last three. 
Formal definition
A subset S of vector space V is called linearly dependent if there exist a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that  Note that the zero on the right is the zero vector, not the number zero. In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...
If such scalars do not exist, then the vectors are said to be linearly independent. This condition can be reformulated as follows: Whenever a1, a2, ..., an are scalars such that
 we have ai = 0 for i = 1, 2, ..., n, i.e. only the trivial solution exists. In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ...
A set is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions.
More generally, let V be a vector space over a field K, and let {vi}i∈I be a family of elements of V. The family is linearly dependent over K if there exists a family {aj}j∈J of elements of K, not all zero, such that In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, an index set is another name for a function domain. ...
 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. In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
Geometric meaning A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered a 2-dimensional vector space (ignoring altitude). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is true, it is not necessary.
In this example the "5 miles north" vector and the "6 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 "7.81 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. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
Note that in this example, any of the three vectors may be described as a linear combination of the other two. While it might be inconvenient, one could describe "6 miles east" in terms of north and northeast. (For example, "Go 5 miles south (mathematically, −5 miles north) and then go 7.81 miles northeast.") Similarly, the north vector is a linear combination of the east and northeast vectors.
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 any location in n-dimensional space.
Example I The vectors (1, 1) and (−3, 2) in R2 are linearly independent.
Proof Let λ1 and λ2 be two real numbers such that In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
 Taking each coordinate alone, this means  Solving for λ1 and λ2, we find that λ1 = 0 and λ2 = 0.
Alternative method using determinants An alternative method uses the fact that n vectors in Rn are linearly dependent if and only if the determinant of the matrix formed by taking the vectors as its columns is zero. ↔ ⇔ ≡ logical symbols representing iff. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...
In this case, the matrix formed by the vectors is
 We may write a linear combination of the columns as  We are interested in whether AΛ = 0 for some nonzero vector Λ. This depends on the determinant of A, which is  Since the determinant is non-zero, the vectors (1, 1) and (−3, 2) are linearly independent. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
When the number of vectors equals the dimension of the vectors, the matrix is square and hence the determinant is defined.
Otherwise, suppose we have m vectors of n coordinates, with m < n. Then A is an n×m matrix and Λ is a column vector with m entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of n equations. Consider the first m rows of A, the first m equations; any solution of the full list of equations must also be true of the reduced list. In fact, if 〈i1,…,im〉 is any list of m rows, then the equation must be true for those rows.
 Furthermore, the reverse is true. That is, we can test whether the m vectors are linearly dependent by testing whether  for all possible lists of m rows. (In case m = n, this requires only one determinant, as above. If m > n, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.
Example II Let V = Rn and consider the following elements in V:  Then e1, e2, ..., en are linearly independent.
Proof Suppose that a1, a2, ..., an are elements of R such that  Since  then ai = 0 for all i in {1, ..., n}.
Example III Let V be the vector space of all functions of a real variable t. Then the functions et and e2t in V are linearly independent. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
Proof Suppose a and b are two real numbers such that - aet + be2t = 0
for all values of t. We need to show that a = 0 and b = 0. In order to do this, we divide through by et (which is never zero) and subtract to obtain - bet = −a
In other words, the function bet must be independent of t, which only occurs when b = 0. It follows that a is also zero.
Example IV The following vectors in R4 are linearly dependent. 
Proof We need to find scalars λ1, λ2 and λ3 such that  Forming the simultaneous equations: In mathematics, simultaneous equations are a set of equations where variables are shared. ...
 we can solve (using for example Gaussian elimination) to obtain: In linear algebra, Gaussian elimination is an algorithm that can be used to determine the solutions of a system of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix. ...
 Since these are nontrivial results, the vectors are linearly dependent.
The projective space of linear dependences A linear dependence among vectors v1, ..., vn is a tuple (a1, ..., an) with n scalar components, not all zero, such that In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...
 If such a linear dependence exists, then the n vectors are linearly dependent. It makes sense to identify two linear dependences if one arises as a non-zero multiple of the other, because in this case the two describe the same linear relationship among the vectors. Under this identification, the set of all linear dependences among v1, ...., vn is a projective space. This article does not cite its references or sources. ...
See also In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence (hence independence structure) that generalizes linear independence in vector spaces. ...
In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ...
External links - MIT Linear Algebra Lecture on Linear Independence at Google Video, from MIT OpenCourseWare
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