Verifying Linear Independence
Learn to verify if given vectors are linearly independent, and if not, find the relationship between the dependent vectors.
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Linear dependence
We defined linear dependence as a collection with linearly dependent vectors if, given a collection of vectors , there exists a combination of scalars with at least one nonzero value such that
There could be infinitely many possible linear combinations, and it’s not possible to check them one by one. We’ve already discussed a few special cases and tricks to check for linear dependence in a combination. This lesson will use elimination as a formal method to solve the vector equation and find the possible combinations.
Procedure
Our goal is to find the combination of scalars , such that
Expanding the vectors, we get
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