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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 **R**^{3} we have the following example.

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.

## Contents

## Formal definition

Let **v**_{1}, **v**_{2}, ..., **v**_{n} be vectors. We say that they are *linearly dependent* if there exist numbers *a*_{1}, *a*_{2}, ..., *a*_{n}, not all equal to zero, such that

Note that the zero on the right is the zero vector, not the number zero.

If such numbers do not exist, then the vectors are said to be *linearly independent*. This condition can be reformulated as follows: Whenever *a*_{1}, *a*_{2}, ..., *a*_{n} are numbers such that

we have *a*_{i} = 0 for *i* = 1, 2, ..., *n*, i.e. *only* the trivial solution exists.

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*. The family is *linearly dependent* over *K* if there exists a family {*a*_{j}}_{j∈J} of elements of *K*, not all zero, such that

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.

A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.

## 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.

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 a location in *n*-dimensional space.

## Example I

The vectors (1, 1) and (−3, 2) in **R**^{2} are linearly independent.

### Proof

Let λ_{1} and λ_{2} be two real numbers such that

Taking each coordinate alone, this means

Solving for λ_{1} and λ_{2}, we find that λ_{1} = 0 and λ_{2} = 0.

### Alternative method using determinants

An alternative method uses the fact that *n* vectors in **R**^{n} are linearly dependent if and only if the determinant of the matrix formed by taking the vectors as its columns is 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 *A*Λ = **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.

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 *A*Λ = **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 〈*i*_{1},…,*i*_{m}〉 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* = **R**^{n} and consider the following elements in *V*:

Then **e**_{1}, **e**_{2}, ..., **e _{n}** are linearly independent.

### Proof

Suppose that *a*_{1}, *a*_{2}, ..., *a _{n}* are elements of

**R**such that

Since

then *a _{i}* = 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 *e ^{t}* and

*e*

^{2t}in

*V*are linearly independent.

### Proof

Suppose *a* and *b* are two real numbers such that

*ae*+^{t}*be*^{2t}= 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 *e*^{t} (which is never zero) and subtract to obtain

*be*= −^{t}*a*

In other words, the function *be*^{t} must be independent of *t*, which only occurs when *b* = 0. It follows that *a* is also zero.

## The projective space of linear dependences

A **linear dependence** among vectors **v**_{1}, ..., **v**_{n} is a tuple (*a*_{1}, ..., *a*_{n}) with *n* scalar components, not all zero, such that

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 **v**_{1}, ...., **v**_{n} is a projective space.

## See also

- orthogonality
- matroid (generalization of the concept)
- Wronskian

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