A set is linearly dependent iff there is a vector which is a linear combination of previous vectors

up:: Linear Dependence

Let be a Vector Space, and (out of which no vector is ).

We seek to prove that

is Linearly Dependent there is some index in for which

Linear dependence implies there is a vector dependent on previous ones

We prove by contrapositive: if for all indexes, , we seek to prove that is Linearly Independent.

Since all vectors are non-null, will be L.I.. By hypothesis, all subsequent vectors will not be in the Span of the previous ones, for which The union of a linearly independent set with a vector outside of its span is also linearly independent. Thus, we can construct

If there is a vector dependent on previous ones, the set is linearly dependent

If there is at least one index such that , then the set will be Linearly Dependent, which means that adding the rest of the vectors will still keep it L.D..¹

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References

• Notas sobre Álgebra Linear

Footnotes

1. Because if is already L.D., we have that is L.D., since . ↩