4. Consider the AR(3) model: y, = 0.216y,-3 + ,1e)-WN(0, 1). (a) Find the solutions (roots) of the autoregressive (characteristic) polynomial. Is this model weakly stationary? (b) If z, = (1 –- B)y,, where B denotes the backward shifts operator, which (ARIMA) model does z, follow? Is z, stationary? %3D (c) If y2019 = 0.6, y2020 = 0.5 and y2021 = 1, find the (mean-square optimal) forecasts of y2022 and y2023- (d) Compute the autocovariances y(0) and y(1) of (y,).

Q: (b) For the matrix Determine: (i) (ii) the characteristic equation the characteristic roots. 4 A = 2…

A:

Q: Consider the data: (zo, f(xo)), (₁, f(x1)), (2, f(x₂)), (#3, f(#3))).\\ The Lagrange basis…

A: Lix=x-x0x-x1·..x-xi-1x-xi+1·...x-xnxi-x0xi-x1·..xi-xi-1xi-xi+1·...xi-xn∴Lixj=0     for i≠j1     for…

Q: 4. Define Let T: M2x2 (R) → P2(R) by T (a b) = (a + b) + (2d) x + ba². c d B = {69). C )·(9)· (9)}…

A: We will find the matrix with respect to the basis.

Q: 4. Consider two 3 × 3 matrices A and B with det(A) = -2 and det(B) = -1. Let T(*) = Až be a linear…

A: solution :-

Q: Let B = {1, t − 8, (t − 8)²} : - and C = {1, t − 5, (t − 5)²} - be two bases for P2. The function…

A:

Q: 2. Suppose that B = 10 1 [000] 10 1 such that TB: R3 -> R3 Determine: (1) ТВ Kernel (TB) Range (TB)…

A: Here given matrix as B=101000101 This matrix is the standard matrix of the linear transformation TB

Q: Let c1 (t) = ei + 7 sin(t)j + t*k and c2 (t) = e-Sti + 5 cos(t)j – 7t*k. (Enter your solution as a…

A: Given: Let c1(t)=e5ti+7sintj+t4k and c2(t)=e-5ti+5cos(t)j-7t4k. To Find:      ddt[c1(t)+c2(t)]

Q: Use the Wronskian and any needed theorems to show that f1 = sin x + 2x2 and f2 = e3x are linearly…

A:

Q: 4. Given two bases (b) = transformation y : R² → R² with respect to the basis (f) and let X, be a…

A: Given two bases b=b1,b2 and f=f1,f2 of ℝ2  where  b1=12, b2=34, f1=11, f2=23 Let Af be a matrix of a…

Q: Find a vector parametric equation r(t) for the line through the points P = (4, -3, 4) and Q = (3,…

A:

Q: Show that T(x, y) = (x - y, y (x - 1)) is not a linear transformation from R² to R².

A:

Q: 1. One application of linear transformations is to solving differential equations. Given a…

A:

Q: assume thatf (2, 3) = 8, fx (2, 3) = 5, fy (2, 3) = 7 Which of (a)–(b) is the linearization of f at…

A: Consider the given: f2,3=8fx2,3=5,fy2,3=7,

Q: Given P₂ = { a+at+at² | ª, ª, a are reals } B= {1-1², t+t², 2-t-t²} and E= {1, t, 1²} [1] Show that…

A: Given:  B=1-t2, t+t2, 2-t-t2 E=1,t,t2 To find: The E coordinates of qt where qB=-123 using change -…

Q: equational-style proof to prove the theorem schema + (B+C) = (((Bv C) → C)) = (((BAC) V T)) (3) Use…

A: Let's prove the given theorem schema using equational-style proof. Want to show:1. Prove :   Will…

Q: V = [X1 5 1;3 X2 2;4 0 6] A. V(:,2) B. diag(V) C. V(1,1) + V(2,1) + V(2,2) + V(3,2) D. 10: X1 : 40…

A:

Q: Ax = b must have a solution for eac In Exercises 33 and 34, T is a linear transformation from R2…

A:

Q: Find the characteristic polynomial of the matrix p(x) = de. = A 0 -2 -2 0 5 -3 -1 0

A: Given matrix is A = -5-2005-1-2-30

Q: 2 - 1 Fit a cubic trend function to the data (- 2,11), (- 1,13), (0,13), (1,12), (2,11). Let po(t) =…

A: Given, The data                           -2, 11, -1, 13, 0, 13, 1, 12, 2, 11 and the t-coordinate…

Q: Let T: R² → R³ and S: R³ → R² be two linear transformations. Suppose the matrix of T is 21 -1 and…

A:

Q: a) f(z) = vz + Vz - 1 b) ƒ(z) = vz² + 9

A:

Q: 9. In this exercise, T: R2 R2 is a function. For each of the following parts, state why T is not…

A: Given function is T:R22 is function we have to prove that why T is not linear in upcoming parts.

Q: (3) Let a = (1, –2, 0), a2 = (0, 1, 2), a3 = (5, –6, k) in R³. %3D %3D (a) Are a1, a2 linearly…

A:

Q: If T1 = a1(d/dt)² + b1 (d/dt) + cqI %3D and T2 = a2(d/dt)² + b2(d/dt) + c2I are linear…

A: Linear transformations do not satisfy the commutative law . Because a linear transformation T is…

Q: Let f(x) = (x – b) Ax + c, where A is symmetric and positive definite matrix, b is a vector and c is…

A: Given that  f(X)=X-bTAX+c, where A is symmetric and positive definite matrix, b is a vector and cis…

Q: Use matrix multiplication to show that the linear transformation represented by A = 59 can be…

A: Given: Linear transformation represented by A=-1001. To Prove: A is the combination of reflection…

Q: T(S)-(S(1)." (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: 4. (a) Are the points u = independent? (1, 2, 3), v = (-1,0, 1) and w = (5, 4, 3) linearly (b) If…

A:

Q: 3. Express the matrix cos sin 0 B = sin 0 0 cos 0 as a successive proper and improper orthogonal…

A: Given: The matrix is B=cosθ-sinθ0sinθcosθ000-1. To find: The orthogonal transformation. The…

Q: Q8) Find the Linearization f(x.y.z) = x² + xy + yz +? at (1,1,2)

A:

Q: 5. Consider V=R². Find the coordinates of = (10, 20) with respect to the ordered basis ß= [(1, 1),…

A: we will find the coordinate vector with respect to the given basis.

Q: 1) Find whether the signals are memoryless or memory? a) y(t) = 5x2(t+2) b) w(t) = (t-9)x(t)

A: a) Given signal is: yt=5x2t+2 Substitute: t=1. y1=5x21+2y1=5x23 So, the output at t=1 depends upon…

Q: [1 The columns of 3 = 0 O Suppose that T is a linear transformation from R³ into R² such that T(e₁)…

A:

Q: 3. Each of J, K, L, M and N is a linear transformation from R' to R?. These functions are given as…

A:

Q: Suppose A is not diagonalizable. If the characteristic equation of A is: -λ +2^² - 2³ = 0 then dim…

A: Given :-  A is not diagonalizable. And Characterstics equation.  Topic :- Matrix.

Q: A 3 T = (l:))-I:1. "(;)) T Consider a linear transformation T from R³ to R2 for which 0 8 (A) - (A)…

A:

Q: If {e", xe", e2a is a basis of solutions of y" + ay" + by' + cy = 0, then a = (A) -2 (В) 2 (С) —4.…

A: