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**Unformatted text preview: **Math 205, Spring 2008 B. Dodson Week 5a: (first part) 2.2 Subspaces/Spanning (continued) 2.3 Independence/Bases ———– Linear Independence/Bases and Dimension Vectors v 1 , v 2 , . . . , v k in a vector space V are linearly dependent if there are scalars c 1 , c 2 , . . . , c k so 0 = c 1 v 1 + c 2 v 2 + ··· + c k v k , with at least one c j = 0 . Such a vector equation is said to be a relation of linear dependence among the v 1 , v 2 , . . . , v k . A collection of vectors for which the only solution of the vector equation is the trivial solution, c 1 = 0 , . . . , c k = 0 is said to be linearly independent. Examples: (1) i, j in R 2 ; and (2) { i, j, k } in R 3 ; are linearly independent. Verification: c 1 i + c 2 j + c 3 k = c 1 (1 , , 0) + c 2 (0 , 1 , 0) + c 3 (0 , , 1) = ( c 1 , c 2 , c 3 ) = (0 , , 0) = 0 , only when c 1 = 0 , c 2 = 0 , c 3 = 0 . 2 Our main objective is the definition of basis....

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