The union of a linearly independent set with a vector outside of its span is also linearly independent

up:: Linear Independence

Let be a Vector Space, and a Linearly Independent set.

Let outside of the Span of . Then is linearly independent as well:

because, if it’s not zero, then we’d contradict our hypothesis ( would be inside ‘s span), since

Thus, since is linearly independent, we must have that .

Thus, is linearly independent.

Corollaries

• A set is linearly dependent iff there is a vector which is a linear combination of previous vectors, which means constructing a Linearly Dependent set comes from including a vector which is a Linear Combination of the previous ones
• By contrapositive of the above, it also follows that A set is linearly independent iff all its finite subsets are also L.I.

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References

• Notas sobre Álgebra Linear