reviewlinearindependencethms

# reviewlinearindependencethms - Review of Linear...

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Unformatted text preview: Review of Linear Independence Theorems Math 108 A: Spring 2010 John Douglas Moore June 4, 2010 A. The Linear Dependence Lemma and Replacement Theorem. Primary goals of this course include learning the notions of linear independence and spanning, and developing the ability to prove theorems from linear algebra that use these concepts. Thus you should be able to reproduce the following definitions: Definition. Suppose that V is a vector space over a field F . A list of vectors ( v 1 ,... v n ) from V is said to span V if v ∈ V ⇒ v = a 1 v 1 + a 2 v 2 + ··· + a n v n , for some choice of a 1 ,a 2 ,...,a n ∈ F . Definition. Suppose that V is a vector space over a field F . A list of vectors ( v 1 ,... v n ) from V is said to be linearly independent if a 1 v 1 + a 2 v 2 + ··· + a n v n = ⇒ a 1 = a 2 = ··· = a n = 0 . The list of vectors is said to be linearly dependent if it is not linearly independent. You should know how to prove: Linear Dependence Lemma. Suppose that ( v 1 ,..., v m ) is a linearly depen- dent list of vectors in a vector space V over a field F , and that v 1 6 = . Then there exists j ∈ { 2 ,...,m } such that v j ∈ Span ( v 1 ,..., v j- 1 ) . Moreover, Span ( v 1 ,..., v j- 1 , v j +1 ,... v m ) = Span ( v 1 ,..., v m ) ....
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reviewlinearindependencethms - Review of Linear...

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