Linear Dependence - ME 210 Applied Mathematics for...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Linear Dependence and Independence of Vectors Given any set of m vectors [x 1 ], [x 2 ], … , [x m ] with the same number of components in each, and any set of m scalars c 1 , c 2 , … , c m , if the linear combination of these vectors is a null vector ¸ i.e., c 1 [x 1 ] + c 2 [x 2 ] + … + c m [x m ] = [0] only when c 1 = c 2 = … = c m = 0 then, [x 1 ], [x 2 ], … , [x m ] are said to be linearly independent . If there is at least one non–zero c i for which the condition is valid, then these vectors are said to be linearly dependent .
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Theorem: m vectors [x 1 ], [x 2 ], … , [x m ] each with n components (n–dimensional vectors), are linearly independent if the augmented matrix, whose column vectors are [x 1 ], [x 2 ], … , [x m ], has rank m; that is [ ] [ ] [ ] [ ] m x x x rank m 2 1 = They are linearly dependent if this rank is less than m.
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This note was uploaded on 07/31/2011 for the course ME 210 taught by Professor Farukarinç during the Spring '09 term at Middle East Technical University.

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Linear Dependence - ME 210 Applied Mathematics for...

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