d_concept2 - Lecture 6: Linear Independence, Spanning,...

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Unformatted text preview: Lecture 6: Linear Independence, Spanning, Basis and Dimension. The homogeneous system A x = can be studied from a different perspective by writing them as vector equations; (1) 1 2- 3 3 5 9 5 9 3 x 1 x 2 x 3 = x 1 1 3 5 + x 2 2 5 9 + x 3 - 3 9 3 = A set of vectors { v 1 ,..., v p } in R n is said to be linearly independent if x 1 v 1 + + x p v p = has only the trivial solution x 1 = = x p = 0. The set { v 1 ,..., v p } is said to be linearly dependent if there are weights 1 ,..., p not all 0, such that 1 v 1 + + p v p = . Ex 1 Determine if v 1 = 1 3 5 , v 2 = 2 5 9 , v 3 = - 3 9 3 are linearly independent? Sol They are linearly dependent if (1) has a nontrivial solution. Augmented: 1 2- 3 0 3 5 9 5 9 3 1 2- 3 0- 1 18- 1 18 1 2- 3 0- 1 18 1 33 0- 1 18 0 Since x 3 is free there are nontrivial solutions x 1 =- 33 x 3 , x 2 =18 x 3 , x 3 is free. If we e.g. let x 3 =1 then x 1 =- 33 and x 2 =18 so we have the linear dependence relation- 33 1 3 5 + 18 2 5 9 + 1 - 3 9 3 = 1 2- 3 3 5 9 5 9 3 - 33 18 1 = The columns of A are linearly independent A x = has only the trivial solution....
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This note was uploaded on 04/08/2012 for the course MATH 340 taught by Professor Davidglenn during the Spring '12 term at Boston Architectural.

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d_concept2 - Lecture 6: Linear Independence, Spanning,...

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