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InnerProd - the dot product on R n is x y =<x y> = y...

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Inner Products give Geometry The dot product is an example of an inner product. If x and y are two real 3-vectors, then x · y = x 1 y 1 + x 2 y 2 + x 3 y 3 . If we think of x and y as (column) vectors in R n , then x · y = x t y = n i =1 x i y i as a matrix multiplication. If x and y are (column) vectors in C n , then x t y is not well-behaved. We might instead use x t y or x t y . These are both perfectly reasonable, equally well-behaved generalizations of the dot product to C n . Before you choose one, I should warn you that there are infinitely many perfectly reasonable generalizations of the dot product. We call them inner products . An inner product is a function that takes two vectors and gives a scalar and which satisfies some properties that makes it “well-behaved”. Specifically, if V is a vector space over the field F , then f : V × V F is an inner product on ( V, F ) if for all x, y, z V and all α F 1. f ( x, x ) > 0 for all x = 0, 2. f ( x, y ) = f ( y, x ) , and 3. f ( αx + y, z ) = αf ( x, z ) + f ( y, z ) . When f is an inner product, we usually denote f ( x, y ) by <x, y> . In this notation,
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Unformatted text preview: the dot product on R n is x · 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 doesn’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 defines a length ( the natural norm for the inner product space) as k x k ≡ √ <x, x>, and the angle, α , between vectors by <x, y> = k x kk y k cos( α ) . You might remember this formula as a theorem from Euclidean geometry; the difference is that here we are defining angles through this formula. Among some immediate consequences are the Cauchy-Schwartz inequality: <x, y><y, x> ≤ <x, x><y, y> the parallelogram identity: k x + y k 2 + k x-y k 2 = 2( k x k 2 + k y k 2 ) , and a geometric interpretation of nullspace, called orthogonality: x ⊥ y ⇔ <x, y> = 0 ....
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