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Linear Algebra

Solve a system of 3 equations.

Solve for x(1), x(2), x(3); 1. 27,954.606 x(1) + 11,969.843 x(2) - 7515.1688 x(3) = 6124.3394 2. 11,969.843 x(1) + 5900.332 x(2) - 3586.4121 x(3) = 3054.3092 3. -7515.1688 x(1) - 3586.4121 x(2) + 2513.4532 x(3) = -1756.4525


Let p be a prime in Z. Define Z(p) = {m/n in rational Q | p does not divide n} i) Show that Z(p) is a subdomain of Q ii) Find the units in Z(p) ,

Linear Algebra : Vector Spaces and Inner Products

1) Let { 1, 2, 2........... n} be a basis of an n dimensional vector space over R and A be n Matrix . Let ( 1, 2, 3............... s) = ( 1, 2, 2........... n) A Prove that dim (span { 1, 2, 3............... s}) = Rank (A). 2) Let V1 be the solution space of x1 +x2 + x3............+xn = 0 let V2 be the solution spac

Injections and Surjections a commutative diagram of groups and that the rows are exact,... being homomorphisms. Prove that (a) if and are surjections and is an injection, then is an injection. (b) if , and are injections, then is an injection. 2. For a group extension {e} B H G {e} Prove that G ~ H/ (B).

Two-part question on a finite dimensional normed linear space

See attachment for question. 1 Suppose that  is a finite dimensional normed linear space. a) Let be a basis for . Define Prove that 1, the closed unit ball in , is compact in (, ) b) Prove that any two norms on  are equivalent.

Abstract Algrbra : Subgroups and Quotients

1. Let G be a group and H be a subgroup of G of index equal to 2. Prove that H G 2. Let (G,?) be a group and H G. Prove that if G/H is a p-group and is a p-group then is a p-group H. Please see the attached file for the fully formatted problems.

Events : Positive and Negative Information

An event F is said to carry negative information about an event E, and we write.... Prove or give counterexamples to the following assertions... (See attachment for full question)

Matrices : Rank

A, B and C are matrices. What are their ranks? A) 1 2 3 4 5 6 B) 1 1 1 1 C) 3 3 7 7 11 11

System of Equations : Software Techniques and an Example

1. Many free software mathematics packages on the Internet will solve a system of equations given the coefficients in the system. Problem: find out which of the four techniques (the Method of Addition, the Method of Substitution, Gauss-Jordan Elimination, and Cramer's Rule) is used in the majority of these types of software p

Practice Questions for Standard Differential Equations (12 Questions)

Please see the attached files for the fully formatted problems. This question is concerned with finding the solution of the first order simultaneous equations where a = -2, b = 8, c = -24, d = 30 (i) Find the particular solutions to the differential equations which satisfy the initial conditions x = 16 and y = 3 at t

Matrices : Row Operations and Echelon Form

1) An augmented matrix of a linear system has been reduced by row operations to the following form. Continue the appropriate row operations and describe the solution set of the original system. Please show every step no matter how minor, use the brackets for each reduction and write out every equation change. Please leave

Prove that R is reflexive.

8. Let R be a relation on a set S such that R is symmetric and transitive and for each x ε S there is an element y ε S such that x R y. Prove that R is an equivalence relation (i.e. prove that R is reflexive)

Rootfinding for Nonlinear Equations: Newton's Method

51. Solve the system: x^2 + xy^3 = 9 3x^2y - y^3 = 4 using Newton's method for nonlinear system. Use each of the initial guesses: (x_0, y_0) = (1.2, 2.5), (-2, 2.5), (-1.2, -2.5), (2, -2.5) Observe which root to which the method converges, the number or iterates required, and the speed of convergence.

Undergrad level Linear Algebra

The reduced row-echelon forms of the augmented matrices of three systems are given in the attachment. How many solutions does each system have? 1. The reduced row-echelon forms of the augmented matrices of three systems are given below. How many solutions does each system have? a. │1 0 2 0│ &#9

Undergrad level Linear Algebra

27. Emile and Gertrude are brother and sister. Emile has twice as many sisters as brothers, and Gertrude has just as many brothers as sisters. How many children are there in this family? Please see attachment for the rest of the questions.

Polynomials, Quadratics, Linear Equations and Word Problems

1.Simplify -i^4: answers a.-1 b.1 c.i d.-i 2.Types of Equations Solve by factoring: x4 - 9x2 = 0. answers a.1, -1, 3, -3 b.0, 3, -3 c.9, -9 d.3, -3 3. Two-Dimensional Coordinate System and Graphs; Find the midpoint of the line segment with endpoints (-4, 8) and (7, 2). a.(3/2, 5) b(-11/2, 3) c.(11/2, -3)

Eigenvalues and Eigenvectors

Can you help me answer question by explaining each step please? Find the eigenvectors and eigenvalues of the matrix A = ( 1 3 0 1 1 1 0 1 1 ) Check that all the eigenvectors, v, and the corresponding eigenvalues, are correct by showing that they satisfy Av=Yv

Linear equations

1. Solve the inequality. Write the solution in interval notation and graph the set on the number line. -2(x - 4) 3x + 1 - 5x 2. Solve the following problem by writing an equation and then solving the equation: You invest $7,200 in two accounts paying 8% and 10% annual interest respectively. At the end of the year, the acco

Linear Algebra : Invertible Matrices

Let A be the real 2x2 matrix [a b] [c d] with bc greater than or equal to 0. Prove there exists a real 2x2 invertible matrix S so that S^-1 A S is either diagonal or of the form [x 1] [0 x] where x is the eigenvalue of A.

Linear Algebra : Diagonalizing Matrices

Let B be an nxn matrix with B^2 = B prove that B is diagonalizable, ie there exists an invertible matrix S so that S^-1 B S is diagonal. (Hint: all eigenvalues of B are either 0 or 1. For each k between 0 and n, consider the case when the nullity of B is k.)

Diagonizable Matrix, Inverse and Nullspace

1) If a Matrix A is diagonizable, must it have an inverse ? if so, is it diagonizable? Can {see attachment} be diagonized, does it have an inverse as well as {see attachment} 2) A is mxn For m<n, is there a vector b such that Ax = b does not have any solution? Any trivial solution for Ax = 0? b) Can say the same for m>n ? A