lectures.notes6-7 - distances Once we know what an inner...

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SECTION 6.7 AN INNER PRODUCT SPACE Lots of vector spaces do not come equipped with a dot or inner product, and consequently in such spaces there are no notions of orthogonality or angle or length or distance or least- squares approximations. For example, the vector space P 7 of all polynomials of degree 7 or less has no naturally defined inner product. To remedy this we first need to decide what properties of inner product are important, then set about defining useful such inner products in spaces like P 7 . These properties are given on page 428 and include essentially the “nice algebra” of our familiar dot product, together with a property that makes sure we can use an inner product to define lengths and
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Unformatted text preview: distances. Once we know what an inner product should be, there are multiple examples of such inner products, including new ones on R n and P 7 . We discuss just one, but it’s probably the most useful and pervasive. The vector space C [0 , 1] consists of all continuous functions defined on the interval [0 , 1]. For two such functions f and g , define the inner product of f and g to be h f, g i = Z 1 f ( t ) g ( t ) dt. EXAMPLES. (1) Compute h f, g i , where f ( t ) = 2 + 3 t and g ( t ) = t + 1. Then compute || f || , the projection of g onto f , and a vector/function h so that h is orthogonal to f . HOMEWORK: SECTION 6.7, #21, 22, 23, 24, 25...
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