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