Set #22 MATH 110 : 2015-16 Due: Friday, 18 March 2016 1. The population of Canada, as determined by the Canadian census, was as follows: year 1991 1996 2001 2006 2011 population (in millions) 27.3 28.8 30.0 31.6 33.5 Let t denote the time measured in years from 1991. (a) Suppose that population (measure in millions) is modeled by the linear function p ptq \" a ` bt. Find the least-squares estimates for the parameters a and b. (b) Suppose that the population (measure in millions) is modeled by the exponential function pe ptq \" cet . Linearize the model and use the least-squares method to estimate the parameters c and . ( 2. Fix n P N and let X :\" 2 : 0 n 1 . Let V :\" CX the complex inner product n space, consisting of all functions from X to C with the inner product xf, gy :\" xPX f pxqgpxq \" n1 f ` 2 ` 2 g n . n \"0 (a) For all j P Z with 0 j n 1, show that the functions wj pxq :\" ejxi are pairwise orthogonal and compute }wj pxq}. (b) For all k P Z with 0 k n 1, consider the function # 1 if x \" 2k n hk pxq :\" . 2k 0 if x n ` Which function in the linear subspace W :\" Span w0 pxq, w1 pxq, . . . , wn1 pxq V best approximates the function hk pxq? (c) For all k P Z with 0 k n 1, calculate }projW phk q hk }. 3. Let V be a nite-dimensional complex inner product space. Show that the adjoint operator on EndpV q has the following four properties. (conjugate-linear) For all S, T P EndpV q and for all c, d P C, we have pcS ` dT q \" c S ` d T . (involutive) (identity) (multiplicative) For all T P EndpV q, we have pT q \" T . For the identity operator I P EndpV q, we have I \" I. For all S, T P EndpV q, we have pST q \" T S . page 1 of 1