| 1. | :with equality if and only if and are linearly dependent.
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| 2. | If the functions are linearly dependent then all generalized Wronskians vanish.
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| 3. | Therefore, d is at least the minimum number of linearly dependent columns.
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| 4. | Are linearly dependent, so that the rank of this larger matrix is still 2.
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| 5. | Indeed, the Slater determinant vanishes if the set { ? i } is linearly dependent.
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| 6. | If such a linear dependence exists, then the " n " vectors are linearly dependent.
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| 7. | A violation of this assumption is perfect multicollinearity, i . e . some explanatory variables are linearly dependent.
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| 8. | Since determinants with linearly dependent rows are equal to 0, one is only left with the last one:
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| 9. | For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent.
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| 10. | Typically, piezoelectric expansion is linearly dependent on applied voltage and a simple subtraction can be used to correct for this effect.
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