Suppose that population measure in millions is modeled by

Question

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	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_l(t) = 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 p_e(t) = ce^λt. Linearize the model and use the least-squares method to estimate the parameters c and λ.

2. Fix n ∈ N and let X := {2πl/n : 0 ≤ l ≤ n-1}. Let V :" C^X the complex inner product space, consisting of all functions from X to C with the inner product

:=_x∈χΣf(x) g(x)‾= _l=0Σ^n-1 f(2Πl/n) {g(2Πl/n)}‾.

(a) For all j ∈ Z with 0 ≤ j ≤ n-1, show that the functions ω_j (x) := e^-jxi are pairwise orthogonal and compute ||ω_j (x)||.

(b) For all k ∈ Z with 0 ≤ k ≤ n-1, consider the function

Which function in the linear subspace W := Span (ω₀(x), ω₁₀(x), . . . , ω_n-1(x)) ⊂ V best approximates the function h_k(x)?

(c) For all k ∈ Z with 0 ≤ k ≤ n-1, calculate ||proj_W (h_k) - h_k||.

3. Let V be a finite-dimensional complex inner product space. Show that the adjoint operator on End(V) has the following four properties.

(conjugate-linear)      For all S, T ∈ End(V) and for all c, d ∈ C, we have

                                  (cS + dT)* = c‾ S* + d‾ T*.

(involutive)                For all T ∈ End(V), we have (T*)* = T.

(identity)                  For the identity operator I ∈ End(V), we have I* =  I.

(multiplicative)           For all S, T ∈ End(V), we have (ST)*=  T* S*.