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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Problem Set 3 Posted Wednesday 8 September 2010. Corrected 14 September. Due Wednesday 15 September 2010, 5pm. 1. [20 points] Determine whether each of the following functions ( Â· , Â· ) determines an inner product on the vector space V . If not, show all the properties of the inner product that are violated. (a) V = C 1 [0 , 1], ( u,v ) = Z 1 u ( x ) v ( x ) dx (b) V = C [0 , 1]: ( u,v ) = Z 1 | u ( x ) || v ( x ) | dx (c) V = C [0 , 1]: ( u,v ) = Z 1 u ( x ) v ( x ) e- x dx (d) V = C 1 [0 , 1]: ( u,v ) = Z 1 u ( x ) v ( x ) dx 2. [20 points] Suppose V is a vector space with an associated inner product. The angle âˆ ( u,v ) between u and v âˆˆ V is defined via the equation ( u,v ) = k u kk v k cos âˆ ( u,v ) . Let V = C [0 , 1] and ( u,v ) = R 1 u ( x ) v ( x ) dx . Compute cos âˆ ( x n ,x m ) between u ( x ) = x n and v ( x ) = x m for nonnegative integers m and n . What happens to âˆ ( x n ,x n +1 ) as n â†’ âˆž ?...
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.
- Fall '09