Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this...
Let f=x3 +x2 + x + 1 in Z2 [x]. Write f as a product of irreducible polynomials in Z2 .
Consider the following linear transformation T: R5 → R3 where T(X1, X2, X3, X4, X5) = (*1-X3+X4, 2X1+X2-X3+2x4, -2X1+3X3-3x4+x5) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...
Problem 4. Consider f(x) = x5+ x4 + 2x3 + 3x2 + 4x + 5 ∈ Q[x] and our goal is to determine if f is irreducible over Q. We compute f(1), f(−1), f(5), f(−5) directly and see that none of them is zero. By the Rational Roots Theorem, f has no root in Q. So if f is reducible over Q, it cannot be factored into the product of a linear polynomial and a quartic polynomial (i.e. polynomial of...
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that f(x, y) dx dy f f(x,y) dy)dr. Jo Jo JoJo
Please explain the solution and write clearly for nu, ber 25. Thanks. 25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
Theoretical Part 1. Consider the problem of computing f(x)dx, where f(x) could be any function. Letting X1, X2 IID ~U[0, 2, define three very simple estimators: ff(0)f(2), i2= f(X1)f(X2), fi3 = f(X1/2) + f((X2+2)/2) (a) (5 points) Is ft an unbiased estimator of u? (b) (5 points) Is i2 an unbiased estimator of ? (c) (5 points) Is 3 an unbiased estimator of ? (d) (10 points) Compute the variance of each of the three estimator when f(x) x Theoretical...
(1 point) This problem will illustrate the divergence theorem by computing the outward flux of the vector field F x, y, z = 2ī + 4j + k across the boundary of the right rectangular prism: 1 sx <5,-2 Sys3,-33z37 oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism...
1) (80pts) Consider the following function f(x)--x5-4x4 + 2x3 + x2-3x + 5 Develop a simple program which will give an iterative solution to the problem f(x)=0 by Newton's algorithm. The solution should display the results on an Excel spreadsheet such as the one given below (the example below is given for a different function). Choose the programming language which suits you most, however the program should be able to read data from an Excel spreadsheet and write the successive...