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Linear Algebra

Cubic polynomial interpolation

1. a) Consider the problem of cubic polynomial interpolation p(xi) = yi, I = 0,1,23 with deg(p) ≤ 3 and x0, x1, x2, x3 distinct. Convert the problem of finding p(x) to another problem involving the solution of a system of linear equations. b) Express the system from (a) in the form Ax = b, i

Linear algebra: finding eigenvalues

Hello. I'm trying to do a principle component analysis of the following file of "good values" - to create a good data signature. I need to find the eigenvectors of each data series. How do I derive the eigenvalues (or eigenvectors) and what are they from this data set? (If you could explain how to do it in matlab, that wo

Systems of Equations Application Word Problem

Stephanie bought eighteen pens, some black and the rest blue, for $8.22. The blue pens cost $0.06 more than the black pens. The number of blue pens that she bought was as close as possible to the number of black pens that she bought. What was the price of the black pens?

Linear Equations, Subsets, Interest and Probability (14 Problems)

1) Which of the following equations describe the same line as the equation 3x+ 4y =5? a. y = (3/4)x + 5 b. 6x + 8y = 5 c. y = (3/4)x + 5/4 d. 5 - 3x - 4y = 0 e. none of the above 2) The equation of the vertical line passing through (-3,5) is a. x = -3 b. x = 5 c. y = -3 d. y = 5 e. none of the above 3)

Hahn Banach Theorem Application

Suppose that is a Banach space over K. A subspace M of is said to be complemented in if there exists a subspace N of such that =M N, that is if , then there exists in M and in N such that , and M N . Prove that each finite dimensional subspace of is complemented in . Hint: Suppose that M is a finite dimens

Cramer's Rule: Solving System of Linear Equations

I must solve the following linear equations using matrix methods. x+y-z=-8 3x-y+z=-4 -x+2y+2z=21 I am trying to understand the method of solving for variables of linear equation by forming them into a matrix and solving for the variables. Please help.

Bounded Linear Operators and Bounded Invertibles

Let = c = C is a continuous function . Let = sup : , for each f in Define T: by (T ( ))(t) = for each t , and For each f in . a) Show that is a bounded linear operator on . b) Compute , For each n in N, and compute . c) Suppose that g . Show that the integral equation

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

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)

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 : 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

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

Linear Algebra : Hermitian Similar Matrices

Suppose A & B are Hermitian matrices and AB=BA, show that A and B are simultaneously diagonalizable, ie, there exists an unitary matrix C so that both C*AC adn C*BC are diagonal.

Eigenfunction Problem

Given y " + ky = 0; y(0)=0 and y'(1)=0; a) Determine the normalized eigenfunction for this problem; b) Use the results in part (a) to express f(x)=x in an eigenfunction expansion, i.e. determine the expansion coefficients (Cn).

Eigenvalues

Calculate the eigenvalues of this matrix: -16 6 60 2 [Note-- you'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues..... (see attached)]

Vectors : Linear Independence

Let A = [ -5, 6, -19] B= [-1, 2, -3] and C= [-2, 2, -8] **they are linearly independent*** The problem is the same as if the vectors were written vertically. If they are linearly dependent, determine a non-trivial linear relation - (a non-trivial relation is three numbers which are not all three zero.) otherwise, if t

Eigenvalues and Eigenvectors

The matrix A = 1 1 0 0 0 0 0 1 1 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and the eigenvectors. Eigenvalue of multiplicity 1 : Associated Eigenvector: Eigenvalue of multiplicity 2 : Associated two linearly independent

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