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**Unformatted text preview: **___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 7-1 Chapter 7. Distance and approximation ♣ We have a clear idea on the length of a vector and a distance (or relation) between two vectors in the Euclidean vector space; we want to extend these concepts to the general vector spaces, while maintaining the basis reasoning behind on those ideas. 1. Inner product spaces A geometrical relation between two vectors in the Euclidean vector space can be defined in terms of the dot product. The inner product is a generalization of the dot product to the general vector spaces and can be used to define length and distance in vector spaces other than n ℜ . In particular, the inner product provides an important tool for dealing with functions. Modern analytical mathematics could not be possible without it. Definition An inner product on a vector space V is an operation (or function) that assigns to every pair of vectors u and v in V a real number < u , v >, s.t. the following properties hold for all vectors u , v , and w in V and all scalar c ; 1. < u , v >=< v , u > 2. < u , v + w >=< u , v >+< u + w > 3. < c u , v >= c < u , v > 4. < u , u > ≥ 0, and < u , u >=0, iff u = . A vector space with an inner product is called an inner product space. ╬ By scalars, we mean the real numbers; i.e. we should refer an inner product space over the real numbers. ╬ In fact, the scalars can be any field. For example, (1) Vector space, n ℜ , with v u v u v u T = • >= < , . (2) Vector space, 2 ℜ , with [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = + >= < 2 1 2 1 2 2 1 1 3 2 3 2 , v v u u v u v u v u . (3) Vector space, n ℜ , with n n n T v u w v u w v u w + + + = >= < ... , 2 2 2 1 1 1 Wv u v u , where ) ,..., ( 1 n w w diag = W with all positive entries. (4) Vector space, n ℜ , with Av u v u T >= < , , where ) ( n n × ∈ A is positive definite. (5) Vector space P 2 with, for 2 2 1 ) ( x a x a a x p + + = and 2 2 1 ) ( x b x b b x q + + = , ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 7-2 2 2 1 1 ) ( ), ( b a b a b a x q x p + + >= < (6) Vector space C [ a,b ] with ∫ >= < b a dx x g x f g f ) ( ) ( , . Theorem 7.1 Let u , v , and w be vectors in an inner product space V , and let c be a scalar. (a) > < + > >=< + < w v w u w v u , , , (b) > < >= < v u v u , , c c (c) , , >= >=< < v u Length, distance, and orthogonality Definition Let u and v be vectors in an inner product space V ....

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