Lecture 18 - Feb 13

# Lecture 18 - Feb 13 - Friday February 13 Lecture 18 Basis...

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Friday February 13 Lecture 18: Basis of a vector space (Refers to section 4.2) Expectations : 1. Define a basis. 2. Recognize that any two bases have the same number of elements. 3. Verify that a set is linearly independent. Verify that a set is a basis. 4. Define dimension of a vector space. 18.1 Definition If { v 1 , v 2 , ...., v k } is both 1. A spanning family of R n (of a subspace W of R n ) and 2. Linearly independent then we say that { v 1 , v 2 , ...., v k } is a basis of R n (of a subspace W of R n ) 18.2 Theorem Any n linearly independent vectors in R n forms a basis of R n . Proof: Suppose { v 1 , v 2 , ...., v n } linearly independent . To show that is a basis it suffices to show that this set spans R n . Let A = [ v 1 v 2 ...., v n ], a square n by n matrix To show that spans R n it suffices to show that Col( A ) = R n . To do this it suffices to show that for any vector v in R n , A x = v is consistent. Since {

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## This note was uploaded on 06/10/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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Lecture 18 - Feb 13 - Friday February 13 Lecture 18 Basis...

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