### Metrics and Euclidean n-Space

Consider the function defined by setting: a) show that the function defines a metric on the Euclidean n-space . Please see the attached file for the fully formatted problems.

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Consider the function defined by setting: a) show that the function defines a metric on the Euclidean n-space . Please see the attached file for the fully formatted problems.

Let m'(A) = inf sum of |M_i| where i is from 1 to infinity, such that A is a subset of M_i. M_i's are disjoint. Is m'(A) = m*(A) ? m*(A) is outer measure.

Please help with the following problem. Provide step by step calculations. Prove that there is a bijection from the open interval (0, 1) to the half-open interval (0, 1].

Let T : V→V be a linear mapping and suppose that x E V is an eigenvector of T corresponding to the eigenvalue λ. Show that x is an eigenvector of T2 corresponding to the eigenvalue λ2. See attached file for full problem description and equations.

Determine the eigenvalues and eigenvectors of each of the following matrices. For each eigenvalue, determine the dimension of the corresponding eigenspace. Please see the attached file for the fully formatted problems.

Prove that ||a|-|b|| ≤ |a-b| for all a,b that are in R.

Show that |b| ≤ a if and only if -a≤b≤a.

Show that (3+square root of 2)^2/3 does not represent a rational number.

Please provide solutions to these two questions (attached). Please show how the subspace satisfies both addition & scalar multiplication! In each of the following exercises 8-17, we will denote by S the set of all vectors x = (x1, x2, x3) E R3 whose coordinates satisfy the given condition. In each case determine whether the

Please provide a semi-detailed response for these *two* questions (attached). --- In each of the following exercises, we shall use the notation f(x) to denote the function x→f(x), x ε R. In each case, V will denote the vector space of all real-valued functions on the real line, with the vector operations defined p

Show that for x,y elements of G are conjugate if they appear symmetrically across the diagonal in the Cayley table of G.

Prove that (a,b,c)=((a,b)c) --- (See attached file for full problem description)

The following are demonstrated: 1) ln(n!) = nlnn + O(n) 2) ln(n!) ~= nlnn - n 3) n! ~= sqrt(2*pi*n) * n^n * e^-n

Find the solution to the given system that satisfies the initial condition x'(t)= [0,2;4,-2]x(t) + [4t;-4t-2] a) x(0)= [4;-5] b) x(2)= [1;1]

Define . Show that the sequence converges. Please see the attached file for the fully formatted problem.

The following two questions I have trouble with: 1. Show that (1- x) n-1. x = (1-x)n-1 - (1-x)n 2. Prove that if ,  are the roots of the equation t² - 2t + 2 then: (x +) n - (x+) n / ( -) = sin n / sin n where cot  =

X + y + 2x = 3 2x - y - 3z = 2 x -2y - z =3 Solve using Gauss method

Solve y = .5x y + 6 = .75(x+6)

Determine q such that the system x + y + 2z = 3 2x -y - 3z = 2 x -2y -qz = 3 Has no solution

1. Solve the following system of simultaneous equations: 6x1 + 4x2 = 40 2x1 + 3x2 = 20 x1 = ??? put your answer in the form x or x.x 2. Max Z = 5x1 + 3x2 Subject to: 6x1 + 2x2 <= 18 15x1 + 20x2 <= 60 x1 , x2 >= 0 Find the optimal profit. Z=? put your answer in the form x.xxx 3. Consider the

The PDE is: xp + yq + p + q -pq = u which gives: p(x+1) + q(y+1) - pq - u = 0 Now, I need to find a complete solution. I have set up my characteristic system to be: dx /x+1-q = dy /y+1-p = du /u-pq = dp /p(pq-u-1) = dq /q(pq-u-1) Help! I cannot solve any of these integrals. I need p = P(x,y,u,a) and

A merchant mixed 10 lb of cinnamon tea with 5 lb of spice tea. The 15 pound mixture sells for $40. A second mixture includes 12 lb of cinn tea and 8 lbs of spice tea. The 20 lbs mixture sells for $54. Find the cost per lb of the cinn tea and the spice tea.

(See attached file for full problem description) --- 1. What is true about the number of solutions to a system of m linear equations in n unknowns if m = n? If m < n? If m > n? Solve each system by the echelon method. 3. 2x + 3y = 10 -3x + y = 18 5. 2x - 3y + z = -5 x + 4y + 2z = 13 5x + 5y + 3z = 14 S

Linear Algebra Matrices (IX) Orthoganal Matrix Show that the matrix A = [-8/9, 4/9 , 1/9; 1/9 , 4/9 , -8/9; 4/9

Find the eigen values and eigen vectors of the matrix A = [9, -1 , 9; 3 , -1 , 3; -7 , 1 , -7] Also find the corresponding eigen spaces for the matrix A. Find a matrix B which reduce the given matrix A to the diagonal form by the transformation B^(-1)AB.

Let T: IR^3 IR^3 , T(x,y,z) = (o,x,2y) - Show that T is nilpotent of index 3 (that is, T^3 = 0 and T^2 different from 0 ) - Find a vector v E IR^3 s.t T^2(v) different from 0 and show that B = { v, T(v), T^2(v) } is linearly independent (so basis) - Find A = [ T ] an

(See attached file for complete problem description) --- Let T: IR^3 IR^3 T(x,y,z) = (y+z, x+z, y+x) B1 = standard basis of IR^3 and B2 the basis B2 = {u1= (1,1,1), u2 = (1, -1,0), u3 = (1,1,-2)} - Find A= [ T ] , B = [ T ] B1

1) Solve the following system of equations that model supply and demand for a product: p - q = 0 ( supply equation ) cp - q = -1 ( demand equation ) where p = price and q = quantity Solve first when c = 0.999 and a second time when c = 1.001. What does the difference in solutions suggest about t

Linear Algebra Matrices Eigen Values and Eigen Vectors(I) Find the characteristic equation of the matrix A = [1, 3, 7; 4, 2, 3; 1, 2, 1] Show that the equation is satisfied by and hence obtain the inverse of the given matrix. The fully formatted problem is in the attached f

Please see attachment. Thank you very much!