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dot product, inner product spaces and coordinates, orthogonal bases, hermitian matrices

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University of Toronto St. George
Martin, Burda

Lecture 13 notes by Y. Burda 1 Reminder about dot product on R n x y n 1 . Recall that on R the dot product of vectors v = . and w = . is xn n dened by the equation v w = x 1 1 ... + x yn n It satises the following properties: Linearity: (c 1 1 c v2 2 w = c v1 1w + c v2 2w Symmetry: v w = w v Positivity: v v > 0 for v 0 2 Inner product spaces n The dot product on R is good for doing geometry in n-dimensional space: measuring lengths, angles, areas and so on. But this is more or less all it is good for. However there are similar notions on other vector spaces, e.g. spaces of functions in analysis or random variables in statistics. These notions allow us to answer questions how close are two functions to each other, how correlated are two events and so on. We will now dene what an inner product is as a generalization of the notion of the dot-product on R : Denition. An inner product on a real vector space V is a way to assign a number (v,w) to any pair of two vectors v,w V so that the following three conditions hold: Linearity: (c v + c v ,w) = c (v ,w) + c (v ,w) 1 1 2 2 1 1 2 2 Symmetry: (v,w) = (w,v) Positivity: (v,v) > 0 for v = 0 Note that symmetry and linearity in the rst argument imply linearity in the second parameter as well: (v,c w + c w ) = c (v,w ) + c (v,w ) 1 1 2 2 1 1 2 2 Example: on the space P (R) o1 degree one polynomial the formula (p(x),q(x)) = p(0)q(0) + p(3)q(3) denes an inner product. 1
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