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Lay_1.7 - 1.7 Linear independence Definition 1 A set of...

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1.7 — Linear independence Definition 1. A set of vectors { a 1 , . . . , a k } is linearly independent if the vector equation x 1 a 1 + · · · + x k a k = 0 (1) has exactly one solution, the trivial solution, x 1 = x 2 = · · · = x k = 0. Definition 2. If there are non-trivial solutions to equation ( 1 ), that is, if there are values x 1 , . . . , x n , that are not all zero and satisfy equation ( 1 ), then the set of vectors { a 1 , . . . , a k } is linearly dependent . Definition 3. If { a 1 , . . . , a k } is a set of linearly dependent vectors, then at least one of the vectors in the set can be expressed as a linear combination of the other vectors in the set (Lay, Chapter 1, Theorem 7).
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