hw4 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 4 Posted...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 4 Posted Wednesday 15 September 2010. Due Wednesday 22 September 2010, 5pm. 1. [20 points] The equation x 1 + x 2 + x 3 = 0 defines a plane in R 3 that passes through the origin. (a) Find two linearly independent vectors in R 3 whose span is this plane. (b) Find the point in this plane closest (in the standard Euclidean norm, k z k = z T z ) to the vector v = 1 0 1 by formulating this as a best approximation problem. (You may use MATLAB to invert a matrix.) 2. [25 points] Recall that a linear operator P is a projection from the vector space V to the vector space V provided P 2 = P , that is, P ( Pf ) = Pf for all f V . Consider V = C [ - 1 , 1] with the usual inner product ( u,v ) = Z 1 - 1 u ( x ) v ( x ) dx, and the two linear operators P e and P o the project a function onto their even and odd parts. That is, ( P e f )( x ) = f ( x ) + f ( - x ) 2 , ( P o f )( x ) = f ( x ) - f ( - x ) 2 . (a) Show that
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hw4 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 4 Posted...

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