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571_5.1 - 5.1 THE SCALAR PRODUCTS IN Rn KIAM HEONG KWA p...

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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 = 0 , the Cauchy-Schwarz inequality is trivially true. Suppose, in the sequel, that x 6 = 0 . The real-valued function f : R R defined by f ( t ) = t 2 k x k 2 - 2 t h x , y i + k y k 2 = k t x - y k 2 Date : July 11, 2011.
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