5 - Linear Dependence

5 - Linear Dependence - Math 1b Practical A theorem on...

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Math 1b Practical — A theorem on linear dependence January 12, 2011 Theorem. If vectors v 1 , v 2 , . . . , v k are linear combinations of vectors u 1 , u 2 , . . . , u r , and if k > r , then v 1 , v 2 , . . . , v k are linearly dependent. We illustrate with an example. Suppose k = 4 and r = 3, and that 3 u 1 + u 2 + 4 u 3 = v 1 2 u 1 + 3 u 3 = v 2 4 u 1 + 2 u 2 u 3 = v 3 5 u 1 u 2 + 2 u 3 = v 4 We may use row operations to get new and equivalent sytems of equations, e.g. we may replace equation 1 by the sum of equation 1 and 7 times equation 2. As usual, we can do this with the matrix of coeficients alone. So start with M = 3 1 4 v 1 2 0 3 v 2 4 2 1 v 3 5 1 2 v 4 or 3 1 4 1 0 0 0 2 0 3 0 1 0 0 4 2 1 0 0 1 0 5 1 2 0 0 0 1 and use pivot operations in the first three columns to make the inital 4 × 3 matrix basic. (We may put linear combinations of the symbols v i in the last column, or we may find it more convenient to add four columns for the coefficients of the v i ’s.) We obtain 1 0 0 (2
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