MATH
312S13FEx-c

# Be the vector space spanned by the two functions e x

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be the vector space spanned by the two functions e x and e x , considered only on the interval [ 1 , 1]. Give V the L 2 inner product: ( f, g ) = integraldisplay 1 1 f ( x ) g ( x ) dx. a) Prove that B = { e x + e x , e x e x } forms an orthogonal basis of V . b) Find the best approximation in V (with respect to the L 2 inner product) of the func- tion g ( x ) = x . [ Hint: think orthogonal projection. Leave your answer in terms of integrals that could be evaluated easily using a computer program. ] B–3. In a large city, a car rental company has three locations: the Airport, the City, and the Suburbs. One has data on which location the cars are returned daily: Rented at Airport: 2% are returned to the City and 25% to the Suburbs. The rest are returned to the Airport. Rented in City : 10% returned to Airport, 10% returned to Suburbs. The rest are returned to the City. Rented in Suburbs: 25% are returned to the Airport and 2% to the city. The rest are returned to the Suburbs. If initially there are 35 cars at the Airport, 150 in the city, and 35 in the suburbs, what is the long-term distribution of the cars?
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