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MATH
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Definitions from Linear independence

All definitions from Linear independence.

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Linear independence

Let SS be a subset of a vector space over FF. The set SS is linearly independent if every finite list of distinct vectors u1,…,un∈Su_1,\ldots,u_n\in S has the property that, for scalars λ1,…,λn∈F\lambda_1,\ldots,\lambda_n\in F,

λ1u1+⋯+λnun=0\lambda_1u_1+\cdots+\lambda_nu_n=0

implies

λ1=⋯=λn=0.\lambda_1=\cdots=\lambda_n=0.

An equation of this form is a relation among the chosen vectors. An independent set is one in which the zero vector has no non-trivial finite representation. An independent set therefore cannot contain the zero vector itself.

  1. [1]:Linear combination
  2. [2]:Span
  3. [3]:Eigenvectors for distinct eigenvalues
  4. [4]:When diagonalisation is possible
  5. [5]:Column independence and nullity
  6. [6]:Basis
  7. 15 more

Linear dependence

A set SS is linearly dependent if some finite list of distinct vectors u1,…,un∈Su_1,\ldots,u_n\in S has scalars, not all zero, such that

λ1u1+⋯+λnun=0.\lambda_1u_1+\cdots+\lambda_nu_n=0.

Dependence is exactly the failure of linear independence.

  1. [1]:Linear independence
  2. [2]:A parameter decides independence
  3. [3]:The homogeneous-system test

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Linear independenceLinear dependence