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Lect3-02-web

Lect3-02-web - MATH 304 Fall 2011 Linear Algebra Homework...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #8 (due Thursday, November 3) All problems are from Leon’s book (8th edition). Section 4.3: 4, 11 Section 5.1: 3c, 5, 7, 17 Section 5.2: 1b, 2a, 4, 9 MATH 304 Linear Algebra Lecture 18: Orthogonal complement (continued). Orthogonal projection. Least squares problems. Euclidean structure Euclidean structure in R n includes: • length of a vector: | x | , • angle between vectors: θ , • dot product: x · y = | x || y | cos θ . A B C θ y x Length and distance Definition. The length of a vector v = ( v 1 , v 2 , . . . , v n ) ∈ R n is bardbl v bardbl = radicalbig v 2 1 + v 2 2 + ··· + v 2 n . The distance between vectors/points x and y is bardbl y − x bardbl . Properties of length: bardbl x bardbl ≥ 0, bardbl x bardbl = 0 only if x = (positivity) bardbl r x bardbl = | r |bardbl x bardbl (homogeneity) bardbl x + y bardbl ≤ bardbl x bardbl + bardbl y bardbl (triangle inequality) Scalar product Definition. The scalar product of vectors x = ( x 1 , x 2 , . . . , x n ) and y = ( y 1 , y 2 , . . . , y n ) is x · y = x 1 y 1 + x 2 y 2 + ··· + x n y n . Properties of scalar product: x · x ≥ 0, x · x = 0 only if x = (positivity) x · y = y · x (symmetry) ( x + y ) · z = x · z + y · z (distributive law) ( r x ) · y = r ( x · y ) (homogeneity) In particular, x · y is a bilinear function (i.e., it is both a linear function of x and a linear function of y ). Angle Cauchy-Schwarz inequality: | x · y | ≤ bardbl x bardblbardbl y bardbl . By the Cauchy-Schwarz inequality, for any nonzero vectors x , y ∈ R n we have cos θ = x · y bardbl x bardblbardbl y bardbl for a unique 0 ≤ θ ≤ π . θ is called the angle between the vectors x and y . The vectors x and y are said to be orthogonal (denoted x ⊥ y ) if x · y = 0 (i.e., if θ = 90 o ). Orthogonality Definition 1. Vectors x , y ∈ R n are said to be orthogonal (denoted x ⊥ y ) if x · y = 0. Definition 2. A vector x ∈ R n is said to be orthogonal to a nonempty set Y ⊂ R n (denoted x ⊥ Y ) if x · y = 0 for any y ∈ Y . Definition 3. Nonempty sets X , Y ⊂ R n are said to be orthogonal (denoted X ⊥ Y ) if x · y = 0 for any x ∈ X and y ∈ Y . Orthogonal complement Definition. Let S ⊂ R n . The orthogonal complement of S , denoted S ⊥ , is the set of all vectors x ∈ R n that are orthogonal to S ....
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Lect3-02-web - MATH 304 Fall 2011 Linear Algebra Homework...

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