571_5.4 - 5.4 INNER PRODUCT SPACES KIAM HEONG KWA pp...

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Unformatted text preview: 5.4 INNER PRODUCT SPACES KIAM HEONG KWA pp. 232-233 An inner product h· , ·i : V × V → R on a linear space V is a function that assigns to each pair of x , y ∈ V a real number h x , y i with the the following properties: (1) h x , x i ≥ for any x ∈ V with equality only if x = . (2) h x , y i = h y , x i for any x , y ∈ V . (3) h α x + β y , z i = α h x , z i + β h y , z i for any x , y , z ∈ V and any α,β ∈ R . It can be shown from property (3) that h , x i = 0 for all x ∈ V . A inner product space is a linear space endowed with an inner product. The following spaces are common instances of inner products spaces. The Euclidean space R n : A standard inner product on R n is the 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 . Let w = ( w 1 ,w 2 , ··· ,w n ) T ∈ R n such that w i > for all i ∈ { 1 , 2 , ··· ,n } . A more general inner product on R n is the weighted scalar product: h x , y i = n X i =1 x i y i w i for any x = ( x 1 ,x 2 , ··· ,x n ) T , y = ( y 1 ,y 2 , ··· ,y n ) T ∈ R n . The vector w is called a weight vector. The space of m × n matrices R m × n : An inner product on R m × n similar to the standard scalar products on Euclidean spaces is given by h A,B i = m X i =1 n X j =1 a ij b ij Date : July 16, 2011. 1 2 KIAM HEONG KWA for any A = ( a ij ) ,B = ( b ij ) ∈ R m × n ....
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571_5.4 - 5.4 INNER PRODUCT SPACES KIAM HEONG KWA pp...

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