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    Linear Algebra

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    Linear system

    Solve the equation. Determine whether it is inconsistent, dependent, or neither. 3x - 2y = 0 9x - 8y = 7

    Linear system

    Solve the linear system. State whether the system is inconsistent, dependent, or neither. x/6 + y/3 = 8 x/4 + y/2 = 12

    System of nonlinear equations

    If a system of nonlinear equations contains one equation whose graph is a circle and another equation whose graph is a line, can the system have exactly one solution? If so, what does the graph of the situation look like?

    Non-linear system

    The following is a non-linear system. Solve it 1/x + 2/y = 3 2/x + 1/y = 4 (Hint: Try a change in variable. Let u = 1/x ; let v = 1/y)

    Linear Algebra with Cubic Roots

    Exercise. IV. This problem is a partial investigation of which n×n matrices over C have cube roots; that is, for which n × n matrices A over C there is an n × n B over C such that A = B3. Since C is algebraically closed, every n × n matrix over C is similar over C to a matrix in Jordan canonical form. A. Suppose that A

    Linear Approximation Function

    I have another question on linear approximation using e. Suppose I want to approximate e^ 0.9. I am assuming that I let f(x) equal the following e^ 0.9 ~ e^0.5 + e^0.5 (0.9-0.5) = 0.4e^0.5 Am I on the right track?

    Complex Matrix, Diagonal Matrix, Left and Right Eigenvectors

    5. Let X^-1 AX = D, where D is a diagonal matrix. (a) Show that the columns of X are right eigenvectors and the conjugate rows of X^-1 are left eigenvectors of A. (b) Let ... be the eigenvalues of A. Show that there are right eigenvectors x1,. . . , x and left eigenvectors y1, . . , yn such that A =... keywords: matrices

    Eigenvalues, eigenfunctions and modified Green's function.

    5.6. (a) Find the eigenvalues and eigenfunctions of ?u"=&#955;u, ?1<x<1; u'(1)?u(1) =0, u'(?1)+u(?1) =0 Show that there is precisely one negative eigenvalue, that zero is an eigenvalue, and that there are infinitely many positive eigenvalues. Show graphically how the eigenvalues are determined. (b) Find the modified Green's f

    Differential Equations

    (See attached file for full problem description) 1) The slope field for the system dx/dt = 2x + 6y dy/dt = 2x - 2y is shown to the right a) determine the type of the equilibrium point at the origin. b) calculate all straight-line solution. 2) show that a matrix of the form A =(a b; -b a) with b!=0 must have complex eig

    eigenvalues and eigenvectors of matrix

    I have some very basic lin algerbra eigenvalue problems. (See attached file for full problem description) 1. Find the eigenvalues and eigenvectors for the projection matrix P = [0.2 0.4 0; 0.4 0.8 0; 0 0 1]; 2. Find the eigenvalues for the permutation matrix P = [0 1 0; 0 0 1; 1 0 0]; 3. Finish the last row to make the mat

    Symplectic Matrix, Eigenvalues and Multiplicity

    A 2n x 2n M is symplectic if where J is the (also 2n x 2n) matrix . Prove that if is an eigenvalue of M , then so is , and that these have the same multiplicity. Show furthermore that, if are eigenvalues of M, and , then the corresponding eigenvectors have the property that Please see the attached file for th

    Pairwise Sequential Voting

    A seventeen-member committee must elect one of four candidates: R, S, T, or W. Their preference schedule as shown below. Which candidate wins under pairwise sequential voting with the predetermined order S, T, W, R? Number of Members Ranking 6 R > S > T > W 5 S > R > T > W 3

    Banach Space and Closed Subspace

    Let I = [a,b] be a finite interval. Show that the space C(I,R^n) of continuous functions from I into R^n is a Banach space with the uniform norm llull = sup l u(t) l where t is in I. (Show that this is a norm and that C(I,R^n) is complete). See attached file. Please be very detailed when answering question.

    Numerical Linear Algebra : Normal Matrices and Eigenpairs

    Combine from JPE, Fall 90 and Spring 97) Let A ∈ C^(n X n) be a normal matrix. a. Prove that A - M is normal for any λ ∈ C. Prove that ||Ax||2 - ||∧*x||2 for all x ∈ C^n b. Prove that ( λ, x) is an eigenpair of A if (λ, x) is an eigenpair of A*. (Hence A and A* have the same eigenvectors.

    Modules, Linear Operators, Characteristic & Minimal Polynomials

    See the attachments. Let F be a field and . Then is an n - dimensional vector space over F. Define a function by . (a) Show that T is a linear operator. (b) Find the characteristic and minimal polynomials for T, with explanation. (For the characteristic polynomial, recall that you will need to choose a basis for

    Partial Differential Equations : Linear and Non-Linear

    A) Solve the equation uxy = x^2 y and its particular solution fur which u(1,y)= cos(y). b) Determine whether each of the following partial differential equations is linear or nonlinear, state the order of the equation, and name the dependent and independent variables. i) ut=9uyy ii) x^2Pzz = z^2Pxx iii) WWrr=pqr iv) sxx + s

    Closed Linear Operator and Banach Space

    Let T be a closed linear operator with domain D(T) in a Banach space X and range R(T) in a normed space Y. If T^-1 exists and is bounded, show that R(T) is closed.

    Difference Equations, Transfer Functions and System Response

    1. When an input x(n) = is applied to a digital filter (which is a linear system), the output is . (a) Find the transfer function of the system, (b) Find the response of the system to a sinusoidal input, Please see the attached file for the fully formatted problems.

    Eigenvalues, Eigenvectors, Characteristic Equation for a Matrix

    2. The time-dependent variables z, y and z satisfy the system of equations Xdot = AX where (a) Show that the characteristic equation for the matrix A may be written as .... (b) In the case of A = 3 determine the corresponding eigenvector. (c) Given that the eigenvectors corresponding to the eigenvalues .. and ... are res

    Linear Programming Proof

    I need the proof of the Linear programming problem attached. --- Consider the LP: Min ct x Subject to Ax &#8805; b, x &#8805; 0. One can convert the problem to an equivalent one with equality constraints by using slack variables. Suppose that the optimal basis for the equality constrained problem is B. Prove t

    Properties of Condition Numbers : Orthogonal Matrices and Eigenvalues

    Please prove the properties of condition numbers attached to this message. Refer to definitions/theorems you used. Also, if you want, have a look at the second file attached, since I believe that you can refer to the previous properties to do 6 to 10. 7. For any orthogonal matrix Q, i2(QA) = k2(AQ) = k2(A) 8. If D= diag(d1,