Linear Algebra

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Show that 2006 is not the difference between two squares.

Show that 2006 is not the difference between two squares. Please see the attached file for the fully formatted problem.

Congruence and Euler's Theorem : Use Euler's Theorem to find the last two digits of 7^1245.

Use Euler's Theorem to find the last two digits of 7^1245. Please see the attached file for the fully formatted problem.

"The mathematical methods in the physical sciences".

Linear algebra questions from the book "The mathematical methods in the physical sciences". See attached file for full problem description.

Solution of linear equation

7(j-5)+8=2(j+5)+5j

Vector Spaces, Linear Transformations, and Eigenvalues / Eigenvectors, Basis and Diagonalizing Matrices

1. Answer the following questions A. Is lambda = 2 an eigenvalue of the following matrix? Why or why not? 3 2 3 8 B. Is lambda - -2 an eigenvalue of the following matrix? Why or why not? 7 3 3 -1 C. Is the following vector an eigenvector of the following matrix? If so, find the eigenvalue. 1 4 -3 1 -3 8 2. Find

Determinants and Cramer's Rule : Row Operations and Effect on Determinant, Computing Determinants and Solving Systems of Equations

1. State the elementary row operation being performed and its effect on the determinant Start with matrix a b c d a. c d a b b. a b kc kd c. a + kc b + kd c d 2. Compute the determinant a. 3 0 4 2 3 2 0 5 -1 b. 1 3 5 2 1 1 3 4 2 c. 3 5 -8 4 0 -2 3 -7 0 0 1 5 0 0 0 2 3. Use row reduction to convert t

Systems of Linear Equations : Row Operations and Solutions

1. Find the augmented matrix for each system of linear equations: a. 5x1 + 7x2 + 8x3 = 3 -2x1 + 4x2 + 9x3 = 3 3x1 - 6x2 + x3 = 1 b. 4x1 + x2 - 7x3 = 6 5x1 + 7x2 + 2x3 = 3 5x1 + 2x2 + 5x3 = 7 c. 3x1 - 2x2 + 2x3 = 7 5x1 + 7x2 + 3x3 = 3 -5x1 + 6x2 - 8x3 = -5 2. Using elementary row operations reduce each of the augm

Vector and Matrix Operations : Row Vectors, Column Vectors, Dot Product and Echelon Form

1. Find the dot product for the following pairs of vectors: a. Row vector = (2 0) Column vector is below 5 18 b. Row vector = (3 9 -4) Column vector is below 3 0 2 c. Row vector = (5 6 7 8) 1 1 1 1 The following matrices will be used in problems 2-3 below: 0 -2 5 A= 3 -4 17 1 2 3 9 7 2 -3

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.

Bijection between open interval and half-open interval

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].

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.

Eigenvalues and Dimension of Corresponding Eigenspace

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 functions for all a,b that are in R.

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.

Linear Subspace Dimension

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

Dimension and 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 point-wise. I

Sum of Newton binomial coefficients with intermittent sign

Sum of Newton binomial coefficients with intermittent sign. See attached file for full problem description.

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

Greatest Common Divisor Proof

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

Approximations to n!, including Stirling's formula

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

Linear Equations

The sum of two numbers is 40. their difference is 18. What are the two numbers?

Solve the followiing system of equations using any method

2x+y=-8 x-y=-4

Solving a System of Equations

Solve the following system of equation using any method. y = 6x + 2 7 = 3x - 7

Inhomogeneous linear system of differential equation

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]

Show That a Sequence Converges

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

Linear functional on N U 0

Let H = l^2(N U 0) (a) Show that if {a_n} is in H, then the power series sum_{n=0}^infty a_n z^n has radius of convergence >= 1. (b) If |b| < 1 and linear functional L: H-->F (F is either the real or the complex field) is defined by L({a_n}) = sum_{n=0}^infty a_n b^n, find the vector h_0 in H such that L(h) = < h, h_0 >

Prove Equalities

See the attached file for the problems.

Working With Matrices

Question: Any matrix B which is formed by the eigen vectors of a matrix A reduces the given matrix A to the diagonal form by the transformation (inverse of B)AB. i.e., (inverse of B)AB = diagonal matrix Please view the attachment to see the fully formatted problem.