Systems of equations and linear dependence
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Testing for linear independence
When a set of equations are linearly dependent then one of the equations is a linear sum of multiples of the others. For example Determine whether the set of equations [Equation goes here - download the original to see it.] Is linearly dependent. If this set is linearly dependent then the third equation would be a linear sum of multiples of the others. Then [Equation goes here - download the original to see it.] For linear dependence there will be non-zero values of that make this equation true. ncoupling, we have [Equation goes here - download the original to see it.] (1)-(2) gives Equation goes here - download the original to see it.] then in (1) Equation goes here - download the original to see it.] Check in (3): LHS = -3x3 + 2x2 = -5 = RHS Which is consistent. Hence this system of equations is linearly dependent since [Equation goes here - download the original to see it.] When equations are linearly dependent, Gaussian row reduction produces a row with zero elements corresponding to the system [Equation goes here - download the original to see it.] We have the augmented matrix [Equation goes here - download the original to see it.] Row reduction results in, for example,: [Equation goes here - download the original to see it.] We can tell that a set of equation is linearly dependent when Gaussian row reduction results in a row with zero elements. Finally, a linearly dependent system will correspond to a matrix with zero determinant. For the syste [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.][Equation goes here - download the original to see it.] With determinant [Equation goes here - download the original to see it.]
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Contents of
Systems of equations and linear dependence
1 Systems of equations and linear dependence
2 Testing for linear independence
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