Lect3-04-web

# Lect3-04-web - MATH 304 Fall 2011 Linear Algebra Homework...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #9 (due Thursday, November 10) All problems are from Leon’s book (8th edition). Section 5.3: 1b, 1c, 3a, 5, 6 Section 5.4: 3b, 7c, 8a, 15b, 30 MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n . Definition. Let V be a vector space. A function α : V → R is called a norm on V if it has the following properties: (i) α ( x ) ≥ 0, α ( x ) = 0 only for x = (positivity) (ii) α ( r x ) = | r | α ( x ) for all r ∈ R (homogeneity) (iii) α ( x + y ) ≤ α ( x ) + α ( y ) (triangle inequality) Notation. The norm of a vector x ∈ V is usually denoted bardbl x bardbl . Different norms on V are distinguished by subscripts, e.g., bardbl x bardbl 1 and bardbl x bardbl 2 . Examples. V = R n , x = ( x 1 , x 2 , . . . , x n ) ∈ R n . • bardbl x bardbl ∞ = max( | x 1 | , | x 2 | , . . . , | x n | ). • bardbl x bardbl p = ( | x 1 | p + | x 2 | p + ··· + | x n | p ) 1 / p , p ≥ 1. Examples. V = C [ a , b ], f : [ a , b ] → R . • bardbl f bardbl ∞ = max a ≤ x ≤ b | f ( x ) | . • bardbl f bardbl p = parenleftbiggintegraldisplay b a | f ( x ) | p dx parenrightbigg 1 / p , p ≥ 1. Normed vector space Definition. A normed vector space is a vector space endowed with a norm. The norm defines a distance function on the normed vector space: dist ( x , y ) = bardbl x − y bardbl . Then we say that a sequence x 1 , x 2 , . . . converges to a vector x if dist ( x , x n ) → 0 as n → ∞ . Also, we say that a vector x is a good approximation of a vector x if dist ( x , x ) is small. Inner product The notion of inner product generalizes the notion of dot product of vectors in R n . Definition. Let V be a vector space. A function β : V × V → R , usually denoted β ( x , y ) = ( x , y ) , is called an inner product on V if it is positive, symmetric, and bilinear. That is, if (i) ( x , x ) ≥ 0, ( x , x ) = 0 only for x = (positivity) (ii) ( x , y ) = ( y , x ) (symmetry) (iii) ( r x , y ) = r ( x , y ) (homogeneity) (iv) ( x + y , z ) = ( x , z ) + ( y , z ) (distributive law) An inner product space is a vector space endowed with an inner product. Examples. V = R n ....
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Lect3-04-web - MATH 304 Fall 2011 Linear Algebra Homework...

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