Linear independence

In the theory of vector spaces, a set of vectors is said to be ' if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such 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 formula_1 from a vector space is said to be "linearly dependent", if there exist scalars formula_2 not all zero, such that
where formula_4 denotes the zero vector.
Notice that if not all of the scalars are zero, then at least one is non-zero, say formula_5, in which case this equation can be written in the form
Thus, formula_7 is shown to be a linear combination of the remaining vectors.
A sequence of vectors formula_8 is said to be "linearly independent" if the equation
can only be satisfied by formula_10 for formula_11 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 formula_12 as a linear combination of its vectors is the trivial representation in which all the scalars formula_13 are zero. Even more concisely, a sequence of vectors is linearly independent if and only if formula_12 can be represented as a linear combination of its vectors in a unique way. 
The alternative definition, that a sequence of vectors is linearly dependent if and only if some vector in that sequence can be written as a linear combination of the other vectors, is only useful when the sequence contains two or more vectors. When the sequence contains no vectors or only one vector, the original definition is used.
Infinite dimensions.
In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. 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" indexed by the set "I". The family is "linearly dependent" over "K" if there exists a non-empty "finite" subset "J" ⊆ "I" and a family } of elements of "K", all non-zero, such that
A set "X" of elements of "V" is "linearly independent" if the corresponding family is linearly independent. Equivalently, a family is dependent if a member is in the closure of the linear span of the rest of the family, i.e., a member is a linear combination of the rest of the family. The trivial case of the empty family must be regarded as linearly independent for theorems to apply.
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 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 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." Although this last statement is "true", it is not necessary.
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.
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 formula_16 is the zero vector formula_4 then the vector formula_16 are necessarily linearly dependent (and consequently, they are not linearly independent). 
To see why, suppose that formula_19 is an index (i.e. an element of formula_20) such that formula_21 Then let formula_22 (alternatively, letting formula_23 be equal any other non-zero scalar will also work) and then let all other scalars be formula_24 (explicitly, this means that for any index formula_25 other than formula_19 (i.e. for formula_27), let formula_28 so that consequently formula_29). 
Simplifying formula_30 gives: 
Because not all scalars are zero (in particular, formula_32), this proves that the vectors formula_16 are linearly dependent. 
As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly "in"dependent. 
Now consider the special case where the sequence of formula_16 has length formula_35 (i.e. the case where formula_36). 
A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero. 
Explicitly, if formula_7 is any vector then the sequence formula_7 (which is a sequence of length formula_35) is linearly dependent if and only if alternatively, the collection formula_7 is linearly independent if and only if formula_41
Linear dependence and independence of two vectors.
This example considers the special case where there are exactly two vector formula_42 and formula_43 from some real or complex vector space. 
The vectors formula_42 and formula_43 are linearly dependent if and only if at least one of the following is true: 
If formula_54 then by setting formula_55 we have formula_56 (this equality holds no matter what the value of formula_43 is), which shows that (1) is true in this particular case. Similarly, if formula_58 then (2) is true because formula_59 
If formula_60 (for instance, if they are both equal to the zero vector formula_4) then "both" (1) and (2) are true (by using formula_62 for both). 
If formula_49 then formula_64 is only possible if formula_65 "and" formula_66; in this case, it is possible to multiply both sides by formula_67 to conclude formula_68 
This shows that if formula_64 and formula_66 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 "in"dependent). 
If formula_49 but instead formula_54 then at least one of formula_48 and formula_43 must be zero. 
Moreover, if exactly one of formula_42 and formula_43 is formula_4 (while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false). 
The vectors formula_42 and formula_43 are linearly "in"dependent if and only if formula_42 is not a scalar multiple of formula_43 "and" formula_43 is not a scalar multiple of formula_42.
Vectors in R2.
Three vectors: Consider the set of vectors formula_84 formula_85 and formula_86 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 "a""i" exist such that formula_92 can be defined in terms of formula_93 and formula_94 Thus, the three vectors are linearly dependent.
Two vectors: Now consider the linear dependence of the two vectors formula_93 and formula_85 and check, 
or
The same row reduction presented above yields,
This shows that formula_100 which means that the vectors "v"1 = (1, 1) and "v"2 = (−3, 2) are linearly independent.
Vectors in R4.
In order to determine if the three vectors in formula_101 
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 "a"i,
where formula_107 can be chosen arbitrarily. Thus, the vectors formula_108 and formula_109 are linearly dependent.
Alternative method using determinants.
An alternative method relies on the fact that formula_110 vectors in formula_111 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 formula_114, which is
Since the determinant is non-zero, the vectors formula_116 and formula_117 are linearly independent.
Otherwise, suppose we have formula_118 vectors of formula_110 coordinates, with formula_120 Then "A" is an "n"×"m" matrix and Λ is a column vector with formula_118 entries, and we are again interested in "A"Λ = 0. As we saw previously, this is equivalent to a list of formula_110 equations. Consider the first formula_118 rows of formula_114, the first formula_118 equations; any solution of the full list of equations must also be true of the reduced list. In fact, if is any list of formula_118 rows, then the equation must be true for those rows.
Furthermore, the reverse is true. That is, we can test whether the formula_118 vectors are linearly dependent by testing whether
for all possible lists of formula_118 rows. (In case formula_131, this requires only one determinant, as above. If formula_132, 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 formula_133
Natural basis vectors.
Let formula_134 and consider the following elements in formula_135, known as the natural basis vectors:
Then formula_137 are linearly independent.
Suppose that formula_138 are real numbers such that
Since
then formula_141 for all formula_142 
Linear independence of functions.
Let formula_135 be the vector space of all differentiable functions of a real variable formula_144. Then the functions formula_145 and formula_146 in formula_135 are linearly independent.
Proof.
Suppose formula_148 and formula_149 are two real numbers such that
Take the first derivative of the above equation:
for values of formula_152 We need to show that formula_153 and formula_154 In order to do this, we subtract the first equation from the second, giving formula_155. Since formula_146 is not zero for some formula_144, formula_158 It follows that formula_153 too. Therefore, according to the definition of linear independence, formula_160 and formula_146 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.
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 formula_118 vectors formula_164 of size formula_110 each, and consider the set of formula_118 augmented vectors formula_167 of size formula_168 each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.
See also: affine space.