Reference no: EM131001076
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 pl(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 pe(t) = ceλt. 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 :" CX the complex inner product space, consisting of all functions from X to C with the inner product
<f, g>:=x∈χΣf(x) g(x)‾= l=0Σn-1 f(2Πl/n) {g(2Πl/n)}‾.
(a) For all j ∈ Z with 0 ≤ j ≤ n-1, show that the functions ωj (x) := e-jxi are pairwise orthogonal and compute ||ωj (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), . . . , ωn-1(x)) ⊂ V best approximates the function hk(x)?
(c) For all k ∈ Z with 0 ≤ k ≤ n-1, calculate ||projW (hk) - hk||.
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*.