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Linear Algebra

System of Equations Application Word Problem

A rectangle window is framed by two rectangular shutters. The window is twice as tall as it is wide. The shutters are as tall as the window, and their widths are 24 cm less than their heights. The combined areas of the shutters total the same as the area of the window. Find the heights. The combined areas of the shutters total t

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

Multivariate Statistics

Suppose that the random variables X1, X2 and X3 have the covariance matrix... (i) Determine the three PC's based on ∑. (ii) What percentage of the total variation is explained by each of the PC's (iii) Calculate the correlation coefficients between the first PC and each of the variables X1, X2 and X3. (iv) What are the

Multivariate Statistics

Consider the following (3x3) matrix A... Find the eigenvalues and eigenvectors of A and hence its spectral decomposition. Hence find... Please see attached for full question.

Ideals

Let R={[a 0] | a,c,d ε Z } and define I={[m 0] | m,n ε Z}. [c d] [n 0] (ii) Is I an ideal in M2(Z)={[a b] | a,b,c,d ε Z}? Why or why not? [c d] I did part (i). So, I only need help on part (ii) Please see attached for

Systems of Ordinary Differential Equations (4 Problems)

Problem 8.1 (Prob. 11, p.251) Solve the following system of equations. { x' = y { y' = -x Problem 8.2 (Prob. 18, p.251) Solve the following system of equations with given initial conditions. { x' = -y { y' = 10x - 7y { x(0) = 2 { y(0) = -7

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

Linear Particle Accelerator : Find Distance Travelled

An electron is accelerated in a linear particle accelerator. At time t = 0, its speed is v0 along the axis of the accelerator, and it is subjected to a force producing a constant acceleration of magnitude a. Select the option which represents the formula most appropriate for finding the distance s that the electron has travel

Linear Equation using Matrices

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.

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

Subdomain

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 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)