Unformatted text preview: the dot product on R n is x Â· y = <x, y> = y t x , while the standard inner product on C n is <x, y> = y t x = y * x . A vector space doesnâ€™t need an inner product, but if it has one, it is an inner product space, and it automatically gets some geometry: an inner product deï¬nes a length ( the natural norm for the inner product space) as k x k â‰¡ âˆš <x, x>, and the angle, Î± , between vectors by <x, y> = k x kk y k cos( Î± ) . You might remember this formula as a theorem from Euclidean geometry; the diï¬€erence is that here we are deï¬ning angles through this formula. Among some immediate consequences are the CauchySchwartz inequality: <x, y><y, x> â‰¤ <x, x><y, y> the parallelogram identity: k x + y k 2 + k xy k 2 = 2( k x k 2 + k y k 2 ) , and a geometric interpretation of nullspace, called orthogonality: x âŠ¥ y â‡” <x, y> = 0 ....
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 Fall '10
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 Dot Product, CN, inner product, Inner product space

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