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Unformatted text preview: The dot product in R n . 1. Dot product in R n : If x = ( x 1 ,...,x n ) and y = ( y 1 ,...,y n ) are points in R n then their dot product is defined by x y = n k =1 x k y k . This is a natural extension to higher dimensions of the familiar dot product from elementary calculus. Observe that the dot product returns a scalar and not another point in R n . More precisely, the dot product is a function : R n R n R . It also can be checked that properties (a)-(d) listed below hold. Because its value is a real number, x y is sometimes called the scalar product of x and y . Another important observation is that x x = k x k 2 . The dot product has some basic properties: (a) x x 0 with equality holding if and only if x = 0, (b) x y = y x , (c) x ( y + z ) = x y + x z and (d) c ( x y ) = ( cx ) y = x ( cy ) for any scalar c . 2. Projections in R n : If x and y are vectors from R n with y 6 = 0 then Proj y x , the projection of x along y...
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- Spring '09