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Lect3-01-web - MATH 304 Fall 2011 Linear Algebra Homework...

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MATH 304, Fall 2011 Linear Algebra
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Homework assignment #8 (due Thursday, November 3) All problems are from Leon’s book. Section 4.3: 4, 11 Section 5.1: 3c, 5, 7, 17 Section 5.2: 1b, 2a, 4, 9
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MATH 304–505 Linear Algebra Part III ( 3.5 weeks): Advanced linear algebra Orthogonality Inner products and norms The Gram-Schmidt orthogonalization process Eigenvalues and eigenvectors Diagonalization Leon’s book : Chapter 5, sections 6.1–6.3
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MATH 304 Linear Algebra Lecture 17: Euclidean structure in R n . Orthogonality. Orthogonal complement.
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Vectors: geometric approach A B A B A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent the same vector if they are of the same length and direction.
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Vectors: geometric approach A B A B v v −→ AB denotes the vector represented by the arrow with tip at B and tail at A . −→ AA is called the zero vector and denoted 0 .
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Vectors: geometric approach A B A B v v If v = −→ AB then −→ BA is called the negative vector of v and denoted v .
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Linear structure: vector addition Given vectors a and b , their sum a + b is defined by the rule −→ AB + −→ BC = −→ AC . That is, choose points A , B , C so that −→ AB = a and −→ BC = b . Then a + b = −→ AC . A B C A B C a b a + b a b a + b
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The difference of the two vectors is defined as a b = a + ( b ). a b b a
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Linear structure: scalar multiplication Let v be a vector and r R . By definition, r v is a vector whose magnitude is | r | times the magnitude of v . The direction of r v coincides with that of v if r > 0. If r < 0 then the directions of r v and v are opposite. v 3 v 2 v
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Beyond linearity: length of a vector The length (or the magnitude ) of a vector −→ AB is the length of the representing segment AB . The length of a vector v is denoted | v | or bardbl v bardbl . Properties of vector length: | x | ≥ 0, | x | = 0 only if x = 0 (positivity) | r x | = | r | | x | (homogeneity) | x + y | ≤ | x | + | y | (triangle inequality) x y x + y
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Beyond linearity: angle between vectors Given nonzero vectors x and y , let A , B , and C be points such that −→ AB = x and −→ AC = y . Then BAC is called the angle between x and y .
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