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section43 - Chapter 4 The Vector Space Rn MAT188H1F Lec03...

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Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.3: Basis and Dimension Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla

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Chapter 4: The Vector Space R n 4.3: Basis and Dimension Basis Deﬁnition: Suppose U is a subspace of R n . A set of vectors B = { X 1 , X 2 , . . . , X k } is called a basis of U if 1. B is linearly independent 2. B is a spanning set for U . That is: a basis is an independent spanning set. If X U , then X = a 1 X 1 + a 2 X 2 + ··· + a k X k , for some scalars a i R , since U = span B . Since B is also independent, these scalars a i are actually unique. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.3: Basis and Dimension That is, suppose X = a 1 X 1 + a 2 X 2 + ··· + a k X k , and X = b 1 X 1 + b 2 X 2 + ··· + b k X k . Then a 1 X 1 + a 2 X 2 + ··· + a k X k = b 1 X 1 + b 2 X 2 + ··· + b k X k ( a 1 - b 1 ) X 1 + ( a 2 - b 2 ) X 2 + ··· + ( a k - b k ) X k = O a 1 - b 1 = 0 , a 2 - b 2 = 0 , . . . , a k - b k = 0 (Why?) a 1 = b 1 , a 2 = b 2 , . . . , a k = b k . So one signiﬁcance of a basis B of U is that every vector in U can be expressed as a linear combination of vectors in B in one and only one way. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
Chapter 4: The Vector Space R n 4.3: Basis and Dimension Example 1 In R 2 : the set B = { ~ i , ~ j } is a basis of R 2 , called the standard basis of R 2 . Recall that ~ i = ± 1 0 ² and ~ j = ± 0 1 ² . B is independent: a ~ i + b ~ j = O ± a b ² = ± 0 0 ² a = 0 and b = 0 . B spans R 2 : Let ~ v = ± x y ² be any vector in R 2 . Then ~ v = x ~ i + y ~ j , as you can easily check. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.3: Basis and Dimension Example 2 In R 3 : the set B = { ~ i , ~ j , ~ k } is a basis of R 3 , called the standard basis of R 3 . Recall that ~ i = 1 0 0 , ~ j = 0 1 0 and ~ k = 0 0 1 . You can check that B is both independent and spanning; it’s very similar to what we did on the previous slide. In general, in R n , if E i = 0 . . . 1

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section43 - Chapter 4 The Vector Space Rn MAT188H1F Lec03...

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