MAT1341-L06-Linear-Independence

MAT1341-L06-Linear-Independence - LINEAR DEPENDENCE AND...

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LINEAR DEPENDENCE AND LINEAR INDEPENDENCE JOS ´ E MALAG ´ ON-L ´ OPEZ Given a finite dimensional vector space V , we would like to know what could be the “minimum” number of vectors required to generate V . We will see that this problem is related to determining in how many ways can a vector in V be written as a linear combination of vectors in a given generator set. Linear Dependence and Linear Independence Roughly , we say that a collection of vectors { v 1 , v 2 , . . . , v m } in a vec- tor space V is linearly dependent if at least one of the vectors in the collection can be written as a linear combination of the other vectors in the collection. Example. The set of vectors - 1 2 0 , 1 - 3 - 1 , 5 - 13 - 3 is linearly dependent since - 2 - 1 2 0 + 3 1 - 3 - 1 = 5 - 13 - 3 . Example. The set of matrices in M 2 × 2 1 2 1 2 , 0 1 2 1 , 2 5 4 3 is linearly dependent since 2 1 2 1 2 + 0 1 2 1 - 2 5 4 3 = 0 0 0 0 1
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Remark. If { v 1 , v 2 , . . ., v m } is a linearly dependent set in V , then one of the vectors in the collection, say v m , is of the form v m = α 1 v 1 + · · · + α m - 1 v m - 1 , where at least one α i is non-zero. Adding - v m on both sides of the equality we get 0 = α 1 v 1 + · · · + α m - 1 v m - 1 - v m .
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