32-innerlecture

32-innerlecture - INNER PRODUCT Math 21b, Fall 2004 RECALL....

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Unformatted text preview: INNER PRODUCT Math 21b, Fall 2004 RECALL. With the dot product in R n , we were able to define angles , length , compute projections or reflections . Especially recall that if ~w 1 , ..., ~w n was an orthonormal set of vectors, then ~v = a 1 ~w 1 + ... + a n ~w n with a i = ~v · ~w i . This was the formula for the orthonormal projection in the case of an orthogonal set. We will aim to do the same for functions. But first we need to define a ”dot product” for functions. THE INNER PRODUCT. For piecewise smooth functions on [- π, π ], we define the inner product h f, g i = 1 π R π- π f ( x ) g ( x ) dx It plays the role of the dot product in R n . It has the same properties as the usual dot product: for example, h f + g, h i = h f, h i + h g, h i or h λf, g i = λ h f, g i . EXAMPLES. • f ( x ) = x 2 and g ( x ) = √ x . Then h f, g i = 1 π R π- π x 3 / 2 dx = 1 π x 5 / 2 2 5 | π- π = 4 5 √ π 3 . • f ( x ) = sin 2 ( x ), g ( x ) = x 3 . Then h f, g i = 1 π R π- π sin 2 ( x ) x 3 dx = ... ? Hold on with the second example, before integrating. It is always a good idea to look for some symmetry. You can immediatly see the answer if you realize that....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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