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# Linear Algebra

### Proof of diagonalizability

Verify: (a) If A is diagonalizable and B is similar to A then B is also diagonalizable. (b) If {see attachment} and x is an eigenvector of A corresponding to an eigenvalue ... {see attachment for complete question

### Homomorphisms

? Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ii) Find the centralizer p in . iii) Prove that for any element a in a g

### Solbing System of Linear Equations

1. Solve by substitution or elimination method: 3x - 2y = 8 -12x + 8y = 32 2. Solve by substitution or elimination method: 7x - 5y = 14 -4x + y = 27 3. Solve by substitution or elimination method: -4x + 3y = 5 12x - 9y = -15 4. A university boo

### Solving Systems of Linear Equations

1. Why do intersecting lines represent a unique solution? Give examples to support your answer. 2. What is the significance of the name 'linear equation' to its graphical representation? 3. The solutions of line m are (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471). The solutions of line n are (3, -9)

### Equations

1.Can you show that, given two equations y = m1x + c1 and y = m2x + c2 where c1 and c2 are different, there is no solution if m1 = m2. Interpret this result graphically. Also show that if c1 = c2 then there will be at least one solution no matter what m1 and m2 are. Interpret this result on a graph. 2.In your reading you have

### Eigenvalues for Linear Algebra Class

Please show all the steps involved. 1. An nxn matrix A is said to be nilpotent if A^k = O (the zero matrix) for some positive integer k. Show that all the eigen values of a nilpotent matrix are O.

### Find a Second Linearly Independent Solution

Please see the attached file for the fully formatted problems. Find a second linearly independent solution given the differential equation & non-trivial solution f.

### Matrix : Convergence, Pseudoinverse and Single Value Decomposition

Only problems #3 &4-a,(without using any software). 3 . Ax = b we consider the iterative scheme .... where the matrix Q is nonsingular. (a) If ... for some subordinate matrix norm, show that the sequence produced by the above scheme converges to the solution of the system for any initial vector x(0). 4. Given singular

### Linear dependence of solutions

One solution to ty"-(t+2)y'+2y=0 is exp(t) Find a second linearly independent solution.

### Solving Simultaneous Linear Equations

3. (a) Solve the following systems of equations i) x + 2 y - z = 2 -3 x - y + z = -3 - x+ 3 y - z = 1 ii) 4 x+ -3 y+ z = - 1 -3 x+ y+ -5 z = 0 -5 x -4 z = 0 iii) x1 + x2 + x3 = 3 -3 x1 -17 x2 + x3 + 2 x 4 = 1 4 x -7 x2 + 8 x3 -5 x4 = 1 -5 x2 -2 x3 + x4 = 1 (b) Find the values of k for which

### Linear Differential Equations

Find the fixed points and sketch trajectories in the phase plane for the system: ... using the phase portrait, examine the behaviour of solutions of this system as t&#8594;&#8734; when they start from (x,y)=(-1,0), and when they start from (x,y)=(-1,-1). Please see attached for full question.

### Convex hull vertices

If A={(-1,-1),(3,-4),(-2,5),(0,3),(2,1),(4,7)}what would be the convexhull(A) expressed as the intersection of a minimum number of closed halfplanes. ALSO, if K is the intersection of the halfspaces: {(x,y,z):x>=0} {(x,y,z):y>=0} {(x,y,z):z>=0} {(x,y,z):x+2y+3z<=6} {(x,y,z):x+3y+2z<=6} {(x,y,z):x<=4} what would the ver

### Isomorphisms and eigenvalues

True or false? Justify your answer in each case (giving a proof or a counterexample): Let T:V-->W be a linear transformation which is an isomorphism. Denote its inverse by T-1 . Suppose that (SYMBOL) is an eigenvalue of T. Then answer a, b, c and d. PLEASE SEE ATTACHMENT FOR COMPLETE QUESTION

### Eigenvalues

Please see the attached file for full problem description. ? Let . Compute the characteristic polynomial and all eigenvalues and all eigenvectors of A. ? True or false? Justify your answer in each case (giving a proof or a counterexample); Let be a linear transformation which is an isomorphism. Denote its inverse by

### Linear Transformation and Matrix

Please help me solve the following linear algebra questions involving linear transformation and matrices. (see attached) ? Let and let . Define a map by sending a vector to . a) let and be the standard basis vectors of V. let , and be the standard basis vectors of W. Find the matrix of T with respect to

### Linear Algebra (Linearly Independent Subset of Vector Space)

In each of the attached cases show that the set U is linearly independent subset of the vector space V. Give, with justification, a basis of V which contains the set U.

### Non Linear System - Second Order System

1// Is it safe to state that for non linear system it's safe to assume/start off with x1 dot = x2? 2// The solutions to x1 dot and x2 dot are provided in the attached. I'm unsure how to arrive at said solutions and require assistance. Thanks in advance.

### Minimax Principle - Intermediate Eigenvalue

Use the minimax principle to show that the intermediate eigenvalue {see attachment} is not positive. *Please see attachment for eigenvalue and hint on how to complete the question. Thanks for your help.

### Vector Subspaces Defined

Suppose that V and W are vector subspaces of Rn. If I define: V + W = {v+w: v belonging to V, w belonging to W} How can I prove that V+W is also a vector subspace of Rn and ALSO how could verify that (for example) <(1,0,1), (-1,0,1)> + <(3,2,1)> = R3 Thanks

### Prove the Transitive Theory

Prove the following theory: 1) R1 is a subset of R2 => All of R3, R1R3 is a subset of R2R3 and 2) R1 is a subset of R2 => All of n, (R1)^n is a subset (R2)^n 3) Suppose R is transitive, then for all of n, R^n is a subset of R.

### Systems of Equations : Matrices

Please give step by step instructions and name each step like triangular form, augmented matrix etc so I know when and what to do and can understand it. We are not using calculator so the steps need to be shown to the solution. 1) Solve the system using elementary row operations on the equations of the augmented matrix. Fol

### Systems of Equations : Triangular Form and Matrices

Please give step by step instructions and name each step like triangular form, augmented matrix etc so I know when and what to do and can understand it. We are not using calculator so the steps need to be shown to the solution. 1) The augmented matrix of a linear system has been transformed by row operations into the form be

### Linear Trend Question

The following linear trend equation was developed for the annual sales of the Jordan Manufacturing Company, Y1 = 500 + 60X (in \$ of dollars). By how much per year and per month are sales increasing?

### Operation Research Random Variables

Basic operation research problem proof (Suppose x solves the problem....Let p,q be random variables with expected (mean) values etc) See attachment for details

### The Linear Diophantine Equation

Find the general solution ( if solution exist) of each of the following linear Diophantine equations: (a) 2x + 3y = 4 (d) 23x + 29y = 25 (b) 17x + 19y = 23 (e) 10x - 8y = 42 (c) 15x + 51y

### Systems of Equations : Gaussian Elimination with Scaled Partial Pivoting

In a linear system 2x + 3y =8 -x + 2y - z =0 3x +2 z=9 I am attempting to solve for x,y, and z using the Gaussian elimination with scaled partial pivoting. I also need to show intermediate matrices and vectors.

### Fermat Numbers

The Fermat numbers are numbers of the form 2 ^2n + 1 = &#934;n . Prove that if n < m , then Φn │ϕ m - 2. The Fermat numbers are numbers of the form 2 ^2n + 1 = (Phi)n . Prove that if n < m , then (Phi)n │(Phi)m - 2.

### Linear independence of vectors & Linear dependence of vectors

Linear dependence of vectors Linear independence of vectors Linear combinations of vectors

### Linear Congruences - The Chinese Remainder Theorem

Find all solutions of each of the systems of congruences:- (a) x is congruent to 1 (mod 2) (d) 4x is congruent to 2 (mod 6) x is congruent to 2 (mod 3)

### Sum of two random variables

Z=sum of dots (uniform on {1,2,3,4,5,6} a. M_z(s)=? b. M_z(s)=M_x(s)^2 why? M_z stands for the mgf of x=sum of dots when two dice are tossed