lecture16 - Differential Equations & Linear Algebra...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 16 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Inner product, norm, orthogonality Contents Inner product, norm, orthogonality Euclidean space R n Inner product We have known how to take the dot product of any two vectors in R n . Now we will generalize this idea to abstract vector spaces. The generalization of dot product is inner product . Definition Let V be a vector space. Then an inner product ( , ) on V is a function with domain consisting of pairs of vectors and range real numbers satisfying the following properties. 1. ( x , x ) ≥ with equality if and only if x = . 2. ( x , y ) = ( y , x ) 3. ( x + y , z ) = ( x , z ) + ( y , z ) 4. ( c x , y ) = ( x , c y ) = c ( x , y ) Example Example Let V = C [0 , 1] be the vector space consisting of all continuous functions on the interval [0 , 1] , with the standard + and * . Then define ( f, g ) = Z 1 f ( t ) g ( t ) dt, ∀ f, g ∈ V. The reason why the above ( , ) is an inner product can be seen by direct checking the four properties. 1. ( f, f ) = R 1 f 2 ( t ) dt ≥ 2. ( f, g ) = R 1 f ( t ) g ( t ) dt = R 1 g ( t ) f ( t ) dt = ( g, f ) 3. ( f + g, h ) = R 1 [ f ( t ) + g ( t )] h ( t ) dt = R 1 f ( t ) h ( t ) dt + R 1 g ( t ) h ( t ) dt = ( f, h ) + ( g, h ) 4. ( cf, g ) = R 1 cf ( t ) g ( t ) dt = c ( f, g ) Exercise...
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This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.

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lecture16 - Differential Equations & Linear Algebra...

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