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Unformatted text preview: 5.1 THE SCALAR PRODUCTS IN R n KIAM HEONG KWA p. 199 The scalar product in R n is defined by Scalar product h x , y i = x T y = n X i =1 x i y i for any x = ( x 1 ,x 2 , ,x n ) T , y = ( y 1 ,y 2 , ,y n ) T R n . Note that h x , y i = x T y = y T x = h y , x i . The Euclidean length of x is given by Euclidean length k x k = p h x , x i = x T x , while the distance between x and y is Distance k x- y k = p h x- y , x- y i = p ( x- y ) T ( x- y ) . The Euclidean length is a norm on R n . MATLAB function. dot( x , y ) generates the scalar product of two column vectors x , y R n . MATLAB function. norm( x ) or norm( x ,2) generates the length of a vector x R n . p. 201 Let x , y R n . Then Cauchy-Schwarz inequality |h x , y i| k x kk y k and the equality holds if and only if x = c y or y = c x for some c R . Proof. In the case x = , the Cauchy-Schwarz inequality is trivially true. Suppose, in the sequel, that x 6 = . The real-valued function f : R...
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