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hw3b - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 3 Posted...

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CAAM 336 · 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 ( · , · ) 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 0 u 0 ( x ) v 0 ( x ) dx (b) V = C [0 , 1]: ( u, v ) = Z 1 0 | u ( x ) || v ( x ) | dx (c) V = C [0 , 1]: ( u, v ) = Z 1 0 u ( x ) v ( x ) e - x dx (d) V = C 1 [0 , 1]: ( u, v ) = Z 1 0 u ( x ) v 0 ( x ) dx 2. [20 points] Suppose V is a vector space with an associated inner product. The angle ( u, v ) between u and v V is defined via the equation ( u, v ) = k u kk v k cos ( u, v ) . Let V = C [0 , 1] and ( u, v ) = R 1 0 u ( x ) v ( x ) dx . Compute cos ( x n , x m ) between u ( x ) = x n and v ( x ) = x m for nonnegative integers m and n . What happens to ( x n , x n +1 ) as n → ∞ ? 3. [25 points] Consider the polynomials φ 1 ( x ) = 1, φ 2 ( x ) = x , and φ 3 ( x ) = 3 x 2 - 1, which form a basis for the set of all quadratic polynomials. These polynomials are orthogonal in C [ - 1 , 1] with the usual inner product ( u, v ) = Z 1 - 1 u ( x ) v ( x ) dx.
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