FACTOID # 29: 73.3% of America's gross operating surplus in motion picture and sound recording industries comes from California.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Linearly independent

In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. For instance, in three-dimensional Euclidean space R3, the three vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) are linearly independent, while (2, −1, 1), (1, 0, 1) and (3, −1, 2) are not (since the third vector is the sum of the first two). Vectors which are not linearly independent are called linearly dependent.

 Contents

Let V be a vector space over a field K. If v1, v2, ..., vn are elements of V, we say that they are linearly dependent over K if there exist elements a1, a2, ..., an in K not all equal to zero such that:

or, more concisely:

(Note that the zero on the right is the zero element in V, not the zero element in K.)

If there do not exist such field elements, then we say that v1, v2, ..., vn are linearly independent. An infinite subset of V is said to be linearly independent if all its finite subsets are linearly independent.

To focus the definition on linear independence, we can say that the vectors v1, v2, ..., vn are linearly independent, if and only if the following condition is satisfied:

Whenever a1, a2, ..., an are elements of K such that:

a1v1 + a2v2 + ... + anvn = 0

then ai = 0 for i = 1, 2, ..., n.

The concept of linear independence is important because a set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.

## The projective space of linear dependences

A linear dependence among vectors v1, ..., vn is a vector (a1, ..., an) 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 v1, ...., vn is a projective space.

## Example I

The vectors (1, 1) and (−3, 2) in R2 are linearly independent.

Proof:

Let a, b be two real numbers such that:

Then:

and
and

Solving for a and b, we find that a = 0 and b = 0.

## 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 Rn such that

Since

then ai = 0 for all i in {1, .., n}.

## Example III: (calculus required)

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.

Proof:

Suppose a and b are two real numbers such that

(1)

for all values of t. We need to show that a = 0 and b = 0. In order to do this, we differentiate both sides of (1) to get

(2)

which also holds for all values of t.

Subtracting the first relation from the second relation, we obtain:

and, by plugging in t = 0, we get b = 0.

From the first relation we then get:

and again for t = 0 we find a = 0.

 Topics in mathematics related to linear algebra Edit (http://en.wikipedia.org/w/wiki.phtml?title=MediaWiki:Space&action=edit)

Results from FactBites:

 PlanetMath: linear independence (143 words) In the case of two vectors, linear independence means that one of these vectors is not a scalar multiple of the other. As an alternate characterization of dependence, we have that a set of of vectors is linearly dependent if and only if some vector in the set lies in the linear span of the other vectors in the set. This is version 22 of linear independence, born on 2001-11-14, modified 2003-02-03.
 Linear independence - Wikipedia, the free encyclopedia (488 words) 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. , the three vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) are linearly independent, while (2, −1, 1), (1, 0, 1) and (3, −1, 2) are not (since the third vector is the sum of the first two). The concept of linear independence is important because a set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.
More results at FactBites »

Share your thoughts, questions and commentary here