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Given a set of vectors we want to determine if they are linearly independent or not (i.e. linear dependent). A set of vectors is linearly independent when the linear combination of the vectors is equal to an all-zero vector only in the trivial case when all combining coefficients are zero.
Thus, to determine if a set of vectors is linearly independent, we just have to construct and solve a homogeneous system of equations. If the only solution is the trivial solution, the vectors are linearly independent. If there are more solutions than just the trivial solution, the vectors are linearly dependent.
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