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 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.

Evaluating linear independence

The zero vector 

If one or more vectors from a given sequence of vectors  is the zero vector  then the vector  are necessarily linearly dependent (and consequently, they are not linearly independent). 
To see why, suppose that  is an index (i.e. an element of ) such that  Then let  (alternatively, letting  be equal any other non-zero scalar will also work) and then let all other scalars be  (explicitly, this means that for any index  other than  (i.e. for ), let  so that consequently ). 
Simplifying  gives: 

Because not all scalars are zero (in particular, ), this proves that the vectors  are linearly dependent.

As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly independent.

Now consider the special case where the sequence of  has length  (i.e. the case where ). 
A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero. 
Explicitly, if  is any vector then the sequence  (which is a sequence of length ) is linearly dependent if and only if  alternatively, the collection  is linearly independent if and only if

Linear dependence and independence of two vectors 

This example considers the special case where there are exactly two vector  and  from some real or complex vector space. 
The vectors  and  are linearly dependent if and only if at least one of the following is true: 
  is a scalar multiple of  (explicitly, this means that there exists a scalar  such that ) or 
  is a scalar multiple of  (explicitly, this means that there exists a scalar  such that ). 
If  then by setting  we have  (this equality holds no matter what the value of  is), which shows that (1) is true in this particular case. Similarly, if  then (2) is true because  
If  (for instance, if they are both equal to the zero vector ) then both (1) and (2) are true (by using  for both).

If  then  is only possible if  and ; in this case, it is possible to multiply both sides by  to conclude  
This shows that if  and  then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly independent). 
If  but instead  then at least one of  and  must be zero. 
Moreover, if exactly one of  and  is  (while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).

The vectors  and  are linearly independent if and only if  is not a scalar multiple of  and  is not a scalar multiple of .

Vectors in R2 
Three vectors:  Consider the set of vectors   and  then the condition for linear dependence seeks a set of non-zero scalars, such that

or

Row reduce this matrix equation by subtracting the first row from the second to obtain,

Continue the row reduction by (i) dividing the second row by 5, and then (ii) multiplying by 3 and adding to the first row, that is

Rearranging this equation allows us to obtain

which shows that non-zero ai exist such that  can be defined in terms of  and   Thus, the three vectors are linearly dependent.

Two vectors:  Now consider the linear dependence of the two vectors  and  and check, 

or

The same row reduction presented above yields,

This shows that  which means that the vectors  and  are linearly independent.

Vectors in R4 
In order to determine if the three vectors in  

are linearly dependent, form the matrix equation,

Row reduce this equation to obtain,

Rearrange to solve for v3 and obtain,

This equation is easily solved to define non-zero ai,

where  can be chosen arbitrarily.  Thus, the vectors  and  are linearly dependent.

Alternative method using determinants 

An alternative method relies on the fact that  vectors in  are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.

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  for some nonzero vector Λ. This depends on the determinant of , which is

Since the determinant is non-zero, the vectors  and  are linearly independent.

Otherwise, suppose we have  vectors of  coordinates, with  Then A is an n×m matrix and Λ is a column vector with  entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of  equations. Consider the first  rows of , the first  equations; any solution of the full list of equations must also be true of the reduced list. In fact, if  is any list of  rows, then the equation must be true for those rows.

Furthermore, the reverse is true. That is, we can test whether the  vectors are linearly dependent by testing whether

for all possible lists of  rows. (In case , this requires only one determinant, as above. If , 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.

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

Natural basis vectors 

Let  and consider the following elements in , known as the natural basis vectors:

Then  are linearly independent.

Linear independence of functions 

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

Proof 
Suppose  and  are two real numbers such that

Take the first derivative of the above equation:

for  values of  We need to show that  and  In order to do this, we subtract the first equation from the second, giving . Since  is not zero for some ,  It follows that  too. Therefore, according to the definition of linear independence,  and  are linearly independent.

Space of linear dependencies 

A linear dependency or linear relation among vectors  is a tuple  with  scalar components such that

If such a linear dependence exists with at least a nonzero component, then the  vectors are linearly dependent. Linear dependencies among  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.

Generalizations

Affine independence

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.

Linearly independent vector subspaces

Two vector subspaces  and  of a vector space  are said to be  if  
More generally, a collection  of subspaces of  are said to be  if  for every index  where  
The vector space  is said to be a  of  if these subspaces are linearly independent and

See also

References

External links 
 
 Linearly Dependent Functions at WolframMathWorld.
 Tutorial and interactive program on Linear Independence.
 Introduction to Linear Independence at KhanAcademy.

Abstract algebra
Linear algebra
Articles containing proofs