Prove that the solution of the discrete least squares problem is given by the orthogonal projection.

Consider the following relations:

R₁ = {(a, b) ∈ R² ∣ a > b}, the greater than relation
R₂ = {(a, b) ∈ R² ∣ a ≥ b}, the greater than or equal to relation
R₃ = {(a, b) ∈ R² ∣ a < b}, the less than relation
R₄ = {(a, b) ∈ R² ∣ a ≤ b}, the less than or equal to relation
R₅ = {(a, b) ∈ R² ∣ a = b}, the equal to relation
R₆ = {(a, b) ∈ R² ∣ a ≠ b}, the unequal to relation

For these relations on the set of real numbers, find

R2∘R1

R2∘R2

R3∘R5

R4∘R1

R5∘R3

R3∘R6

R4∘R6

R6∘R6

Use your graphing calculator to find the solutions to the following equation for

0° ≤ θ < 360°

by defining the left side and right side of the equation as functions and then finding the intersection points of their graphs. Make sure your calculator is set to degree mode. (Round your answers to one decimal place. Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

3 sin² θ + 1 = 5 sin θ

Use Laplace transforms to solve the initial value problem:

y''' −y' + t = 0,

y(0) = 0,

y'(0) = 0,

y''(0) = 0.

Did any of your previous assignments involve math calculations for premium and or salary analysis and adjustments? Please provide detailed example(s).

Derive Open Newton's Cotes Formula for n=6 ?

Derive Closed Newton-Cotes Formulas for n=5 and n=4 ?

Let A = 0 2 0

1 0 2

0 1 0 .

(a) Find the eigenvalues of A and bases of the corresponding eigenspaces.

(b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line.

(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist.

(d) Write down explicitly a diagonalizing matrix S, and a diagonal matrix Λ such that S −1AS = Λ; A = SΛS −1 . or explain why A is not diagonalizable.

Calculate the half of the ratio of perimeter of regular apeirogon to the distance of the centre of gravity to the edges of this regular apeirogon, find the 1288. 1289. 1290. and 1291. decimal places.

Compute the projection of y = (1, 2, 2, 2, 1)  onto span (x1, x2) where

x1 =(1, 1, 1, 1, 1)   x2 =(4, 1, 0, 1, 4)

The inner product to use is the usual dot product. (This will compute a best-fitting function that is quadratic with no linear term, fitting to the data (−2, 1),(−1, 2),(0, 2),(1, 2),(2, 1).)

Define T: P2 → P2 by T(a0 + a1x + a2x2) = (−3a1 + 5a2) + (−4a0 + 4a1 − 10a2)x + 5a2x2. Find the eigenvalues. (Enter your answers from smallest to largest.) (λ1, λ2, λ3) = −2,5,6 Correct: Your answer is correct. Find the corresponding coordinate eigenvectors of T relative to the standard basis {1, x, x2}. x1 = −5,10,0 Incorrect: Your answer is incorrect. x2 = − 5 2​,5,1 Incorrect: Your answer is incorrect. x3 = − 1 2​,1,0 Incorrect: Your answer is incorrect.

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =

at time t=0, a particle is located at the point(4,8,7). it travels in a straight line to the point (7,1,6), has speed 7 at (4,8,7) and constant acceleration 3i-7j-k. Find an equation for the position vector r(t) of the particle at time