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Unformatted text preview: LINEAR DEPENDENCE AND LINEAR INDEPENDENCE JOS ´ E MALAG ´ ONL ´ 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 , 1 3 1 , 5 13 3 is linearly dependent since 2  1 2 + 3 1 3 1 = 5 13 3 . Example. The set of matrices in M 2 × 2 1 2 1 2 , 1 2 1 , 2 5 4 3 is linearly dependent since 2 1 2 1 2 + 1 2 1 2 5 4 3 = 1 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 nonzero. Adding v m on both sides of the equality we get...
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This note was uploaded on 01/22/2011 for the course MAT 1341 taught by Professor Josemalagonlopez during the Fall '10 term at University of Ottawa.
 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra, Vectors, Linear Independence, Vector Space

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