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# 107 - Given a set of vectors how to determine if there are...

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1 Given a set of vectors, how to determine if there are any vectors that are linear combinations of other vectors? Idea: in { u 1 , , u i , , u k }, if is u i a linear combination of other vectors, then there exists scalars c 1 , , c i , , c k , not all zero , such that c 1 u 1 + " + c i u i + " + c k u k = 0 . (at least c i = 1 ) ( L.D. ) ( L.I. ) Property: Any finite set S = { 0 , u 1 , u 2 , , u k } that contains the zero vector is L.D., since 1· 0 + 0· u 1 + 0· u 2 + " +0 · u k = 0 . Condition for L.D.: Example: L.D. or L.I.? and which element(s), if any, can be linearly combined by others? A = [ " ]

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2 the augmented matrix of A x = 0 is reduced row echelon form the general solution of A x = 0 is setting x 3 = 1 leads to Conclusion: S is L.D., and the last vector is not a linear combination of others. In general, the set is L.I. if and only if there is not any free variable. always zero column, redundant. (g) There is a pivot position in each column of A . Proof (a) (f): by definition, as noted.
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107 - Given a set of vectors how to determine if there are...

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