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Unformatted text preview: 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 = ~ . 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 kk ~v k ( k ~u kk ~v k- ~u ~v ) , establishing (applying this to both ~u and- ~u ) the Cauchy-Schwarz Inequality | ~u ~v | 6 k ~u kk ~v k . This guarantees that the ratio ~u ~v k ~u kk ~v k falls in the interval [- 1 , 1], hence is uniquely cos( ) for some in the interval [0 , ]; the angle between two vectors ~u,~v is = arccos ~u ~v k ~u kk ~v k [0 , ] . 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 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 ....
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This note was uploaded on 09/25/2009 for the course APMA 3080 taught by Professor Pindera during the Spring '09 term at UVA.
- Spring '09