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Week3.5Syllabus

Week3.5Syllabus - Week 3.5 Syllabus The Metric Structure of...

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Week 3.5 Syllabus: The Metric Structure of R n , Applications, Matrix Operations 1 The Metric Structure R n R n (ro vectors for ease of typography, similarly for column vectors R n ) has a dot product on vectors defined as ~u ~v = n i =1 u i v i R , i.e., multiply the coefficients term by term but then add the results to form a single scalar . It satisfies the following basic rules for all vectors ~u, ~u i ,~v V and scalars c R : (1) ( symmetry ) ~u ~v = ~v ~u (2) ( additivity ) ( ~u 1 + ~u 2 ) ~v = u 1 ~v + u 2 ~v (3) ( homogeneity ) ( c~u ) ~v = c ( ~u ~v ) (4) ( positive-definiteness ) ~u ~u > 0 for all ~u 6 = ~ 0 . These 4 rules will become the definition for an abstract inner product on any vector space. 1.1 The Norm From the dot product we can derive the the concepts of norm of a vector k ~v k := ~v ~v = p x 2 1 + · · · x 2 n 0 satisfying k ~u k = 0 iff ~u = ~ 0 and k c~u k = | c | k ~u k . Any nonzero vector ~v can be normalized to give a (normal) unit vector ~u with k ~u k = 1 pointing a length 1 in the same direction via ~u = 1 k ~v k ~v (sloppily but conveniently written ~v k ~v k ). Remember that because the norm involves a square root, itis usually more convenient for calculations to deal with k ~v k 2 = ~v ~v, since dot products are bilinear but square roots are not. 1.2 The Cauchy-Schwarz Inequality Expanding out the positive-definiteness ~x ~x 0 for x := k ~u k ~v -k ~v k ~u yields 2 k ~u k k ~v k ( k ~u k k ~v k- ~u ~v ) 0 , establishing (applying this to both ~u and - ~u ) the Cauchy-Schwarz Inequality | ~u ~v | 6 k ~u k k ~v k . This guarantees that the ratio ~u ~v k ~u k k ~v k falls in the interval [ - 1 , 1], hence is uniquely cos( θ ) for some θ in the interval [0 , π ]; the angle θ between two vectors ~u,~v is θ = arccos ~u ~v k ~u k k ~v k · [0 , π ] . The most important angle is pi 2 (a.k.a 90 ): orthogonality (a.k.a. perpendicularity) ~u ~v of two vectors means ~u ~v = 0. The CSI implies the triangle inequality k ~u + ~v k ≤ k ~u k + k ~v k (by squaring both sides). This will be very important in measuring distances between vectors (especially functions) 1.3 Distance We have a metric concept of distance between two vectors d ( ~u,~v ) := k ~u - ~v k , satisfying d ( x, x ) = 0 ⇐⇒ x = 0 , d ( x, y ) = d ( y, x ), and the triangle inequality d ( x, z ) 6 d ( x, y ) + 1
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d ( y, z ) [this follows from the triangle inequality for vectors with u = x - y, v = y - z, u + v = x - z ]. The Pythagorean Theorem says that if ~u,~v are orthogonal then k ~u + ~v k 2 = k ~u k 2 + k ~v k 2 . From these we can (in some other course) study calculus in R n . 2 Applications The text introduces several instances where practical problems lead to systems of linear equations. There is no one set of applications of interest to all students, and the applications given in the book are extremely elementary. Many applications involve ”network analysis” where several systems are interconnected in a linear way, and the problem is to analyze the network as a whole.
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