This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ?...
View
Full
Document
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
 Tompson

Click to edit the document details