Basic Examples (4) 

Test some two-dimensional vectors for linear independence:

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Test some three-dimensional vectors for linear independence:

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This set of vectors is linearly dependent:

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Confirm that the third vector can be written as a linear combination of the first two:

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LinearlyIndependent works with any number of vectors of any dimension:

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Scope (3) 

For vectors with symbolic parameters, LinearlyIndependent may return a ConditionalExpression:

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A True/False result may be obtained by giving values to the parameters:

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LinearlyIndependent accepts vectors with complex components:

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Properties and Relations (6) 

A set of vectors is linearly independent if and only if the rank of the row matrix composed of the vectors equals the length of the vectors:

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A set of vectors is linearly independent if and only if the rank of the row matrix composed of the vectors has a zero-dimensional null space:

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Or, alternatively:

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A set of vectors is linearly independent if and only if the rank of the row matrix composed of the vectors has a nonzero determinant:

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A set of vectors is linearly independent if and only if its row-reduced form has a no zeros along its diagonal:

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The zero vector is linearly dependent on every other vector:

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LinearlyIndependent will not evaluate if the vectors do not all have the same length:

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