linearly independent
Let V be a vector space over a field F. We say that v1,…,vk∈V are linearly dependent if there exist scalars λ1,…,λk∈F, not all zero, such that
λ1v1+⋯+λkvk=0. |
If no such scalars exist, then we say that the vectors are linearly independent. More generally, we say that a (possibly infinite) subset S⊂V is linearly independent if all finite subsets of S are linearly independent.
In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following.
Proposition 1.
Let S⊂V be a subset of a vector space. Then, S is linearly dependent if and only if there exists a v∈S such that v can be expressed as a linear combination of the vectors in the set S\{v} (all the vectors in S other than v (http://planetmath.org/SetDifference)).
Remark. Linear independence can be defined more generally for modules over rings: if M is a (left) module over a ring R. A subset S of M is linearly independent if whenever r1m1+⋯+rnmn=0 for ri∈R and mi∈M, then r1=⋯=rn=0.
Title | linearly independent |
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Canonical name | LinearlyIndependent |
Date of creation | 2013-03-22 11:58:40 |
Last modified on | 2013-03-22 11:58:40 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 30 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A03 |
Synonym | linear independence |
Defines | linearly dependent |
Defines | linear dependence |