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Linear Algebra

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.

Proof : Outer Measure

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.

Eigenvector of a linear mapping

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.

Dimension, Linear Dependence

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

Prove Equalities

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  =

Linear Algebra

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

PDE : Complete solution to a non-linear equation.

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

Find the eigen values and eigen vectors of the matrix

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.

Characteristic equation of a matrix and inverse of a matrix

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