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Encyclopedia > Linearly dependent

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

Definition

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.


See also

Topics in mathematics related to linear algebra

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Vectors | Vector spaces | Linear span | Linear transformation | Linear independence | Linear combination | Basis | Column space | Row space | Dual space | Orthogonality | Eigenvector | Eigenvalue | Least squares regressions | Outer product | Cross product | Dot product | Transpose | Matrix decomposition


  Results from FactBites:
 
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.
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.
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.
PlanetMath: linear independence (143 words)
Otherwise, if this conditions fails, the vectors are said to be linearly dependent.
Furthermore, an infinite set of vectors is linearly independent if all finite subsets are linearly independent.
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.
  More results at FactBites »

 
 

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