3. Let G1 and G2 be groups with identity elements e1 and e2, respectively. a. Prove that G1×{e2} is a normal subgroup of G1×G2. (You do not need to prove that G1×{e2} is a subgroup, since this follows from a previous homework problem, just that it is normal in G1 ×G2.) b. Prove that (G1 ×G2)/(G1 ×{e2}) ∼= G2

A tank contains 2800 L of pure water. A solution that contains 001 kg of sugar per liter enters a tank at the rate 9 L/min The solution is mixed and drains from the tank at the same rate. a) How much sugar is in the tank initially? b) Find the amount of sugar in the tank after t minutes. amount = (kg) (your answer should be a function of t) c) Find the concentration of sugar in the solution in the tank after 45 minutes. concentration = (kg/L)

Prove or disprove if B is a proper subset of A and there is a bijection from A to B then A is infinite

How many positive integers less than 1,000,000 have exactly one digit

that is 7 and the product of this digit (7) with the sum of other digits is

between 50 and 65?

A professor wants to determine if there is a difference between students' pre and post test after a semester of learning.

1. What could be the research question for this problem?

2. What could be the null hypothesis and alternative hypothesis for this problem?

3. Looking at the mean of both pro and post test, on the end-results below, does it REJECT or FAIL TO REJECT the null hypothesis and why?

The results of the test showed this following:

B= {1,x,x^2}

B’={1-x^2, x-x^2, 1+2x-x^2}

T(a+bx+cx^2) = 3a+b+c+(2a+4b+2c)x + (-a-b+c)x^2

a) by direct calculation , compute [P]_B’ , p=7-x+2x^2

b) using basis B={1,x,x^2} , compute [T]_B

c_ compute [T]_B’

T(1+2x)=1+x-x^2

T(1-x^2)=2-x

T(1-2x+x^2)=3x-2x^2

a)compute T(-6x+3x^2)

b) find basis for N(T), null space of T

c) compute rank of T and find basis of R(T)

Find a basis of U = span {(1,1,2,3), (2,4,1,0), (1,5,-4,-9)}

Find an injective morphism Z2 × Z2 → S4. Is it possible to find an inclusion into S3?

Given the vectors u1 = (2, −1, 3) and u2 = (1, 2, 2) find a third vector u3 in R3 such that

(a) {u1, u2, u3} spans R3
(b) {u1, u2, u3} does not span R3

Suppose a dog is chasing a rabbit along the y-axis, and the coordinate of the rabbit's position at time t is given by (1/3) t^2 +4t. The dog's speed is equal to one half the distance between the two animals and the dog's coordinate at t=0 is y=-5.

a. Determine a differential equation for the position of the dog

b. Justify whether the dog will ever catch the rabbit?

how can I change the Gauss-Seidel method to SOR method code in Matlab?

The question has shows that In implementing SOR method in MATLAB, one should not calculate Tw and cw by formulas Tw = (D -wL)^(-1)[(1-w)D+wU)] and Cw = w(D-wL)^(-1)b , where w stands for omega and using MATLAB's built-in inv function, since this function requires O(n^3) flops and therefore the
whole matter loses its point.

I have tried for many times but I can't get the correct answers. Please help Thanks :)

Execute code:

function x = GaussSeidel(A,b,x0,tol,kmax,output)
% This function returns an approximate solution of a
% linear system A*x=b, obtained by the Gauss-Seidel method.
%
% x0 is an initial approximation,
% tol is tolerance,
% kmax is the maximum number of iterations,
% output is a parameter which regulates displays:
% 0 - no display,
% 1 - some display,
% 2 - detailed display.
%
% Input: A, n by n matrix,
% b, n by 1 column,
% x0, initial approximation,
% tol, tolerance,
% kmax, maximum number of iterations,
% output, a parameter which regulates displays.
%
% Output: x, a solution.

[m,n]=size(A);
if m~=n,
error('A is not square');
end
m=length(b);
if m~=n,
error('Dimensions of A and B do not agree');
end
% Write A as D-L-U
[D,L,U] = DLU_decomposition(A);
% Calculate the matrix T and the vector c
T = L+U;
c=b;
for i=1:n,
T(i,:) = T(i,:)/D(i,i); % T = D^(-1)*(L+U)
c(i) = b(i)/D(i,i); % c = D^(-1)*b
end
% Calculate x
x=x0; % initial approximation
if output >= 2
disp(' k TOL x1 x2');
end
for k=1:kmax,
xprev=x;
for i=1:n
x(i)=c(i);
for j=1:n
if j~=i
x(i)=x(i)+T(i,j)*x(j);
end
end
end
TOL=norm(x-xprev,inf);
if output >= 2
out=[k, TOL, x(1), x(2)];
disp(out);
end
if TOL<tol,
if output >= 1
disp('The Gauss-Seidel method has converged.');
end
break
end
end
if output >= 1
if TOL>=tol
disp('The Gauss-Seidel method did not converge.');
end
s=sprintf('The norm of residual vector A*x-b is %e.',norm(A*x-b));
disp(s);
end
end