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    Linear Algebra

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    Subdomain Proof

    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

    .....is 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 Algebra : 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

    Psychology Laboratory Conducting Dream Research

    10. A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3 sets of identical twins are to be assigned to these 6 beds so that each set of twins sleeps in different beds in the same room, how many assignments are possible?

    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

    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: Vectors and Planes

    1) Let v_1 = 1 v_2 = -3 and y = h. [ 0 ] [ 8 ] [-5] [-2 ] [ 8 ] [-3] For what value of h is y in the plane generated by v_1 and v_2? 2) Compute the product why (a) the definitions and (b) the raw vector rule. 8 3 -4 * 1 5 1 2

    Matrices: Existence and Uniqueness of Solutions

    1. Choose h and k such that the system has (a) no solution, (b) a unique solution and (c) many solutions. a) x_1 + hx_2 = 2 4x_1 + 8x_2 = k b) x_1 + 3x_2 = 2 3x_1 + hx_2 = k 2. Explain existence and uniqueness and give examples using a matrix.

    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.

    Reduced row-echelon forms of the augmented matrices

    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