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Linear Algebra

Linear Equation and System of Inequalities

A target heart rate T that is half a person's maximum heart rate is given by T=110 - 1/2A where A is a person's age. (a) What is T for a person 30 years old? 50 years old? (b) Sketch a graph of the system of inequalities. T > 110 - 1/2A T < 220 - A Assume that A is between 20 and 60. (c) Interpret your grap

Solve a system of linear equations with 3 variables

Cost of CDs : The accompanying table shows the total cost of purchasing combinations of differently priced CDs. The types of CDs are labeled A, B, and C. A B C Total Cost 1 1 1 $37 3 2 1 $69 1 1 4 $82 (a) Let x be the cost of a CD of type A, y be the cost of a CD

Linear Algebra, Vector Space and Mapping

See the attached file. Let Beta = {x_1, ..., x_n} be a basis for a vector space V, and let P be the mapping P((a_1)(x_1) + ... + (a_n)(x_n)) = (a_1)(x_1) + ... + (a_k)(x_k). a) Show that Ker(P) = Span({x_k+1, ..., x_n}) and Im(P) = Span ({x_1, ..., x_k}) b) Show that P^2 = P c) Show conversely that if P:V --> V is an

Question about Inequalities and Equations

1.) How many solution sets do systems of linear inequalities have? Must solutions to systems of linear inequalities satisfy both inequalities? In what case might they not? Provide an example and a reference. 2.) Do the equations x = 4y + 1 and x = 4y - 1 have the same solution? How might you explain your answer to someone w

Find the spectral decomposition

** Please see the attached file for the complete solution ** 8) Find the spectral decomposition of A = (please see the attached file) 9) Identify and sketch the conic section given by 3x^2 + 2xy + 3y^2 - 8 = 0

Equations

The techniques for solving linear equations and linear inequalities are similar, yet different. Explain and give an example of both a linear equation and a linear inequality that demonstrates this difference. 1.) Solve and check the linear equation. 5x - 5 = 30 A) {30} B) {34} C) {11} D) {7} 2.) Solve and check th

Algebra Problems & Solving Linear Equations

When solving a linear equation in one variable, the objective is to isolate the variable on one side of the equation. What does that mean? Give an example. 1. Add. (9a3 + 3a2) + (5a3 + 6a2) A) 14a6 + 9a4 B) 23a5 C) 14a3 + 9a2 D) 23a10 2. Add. (-9 + 3n6 + 3n5) + (2n6 + 5n5 + 6) A) 5n6 + 8n5 - 3 B) 10n11 C)

Solving linear systems

To be efficient, under which conditions would you use graphing elimination or substitution? Provide an example. The three methods of solving linear systems covered are substitution, elimination, and graphing. There are examples posted on the solution field and in the attachment.

Rational and Irrational Number Proof

Prove or disprove each of the following: a. The sum of a rational number and an irrational number is an irrational number. b. The product of two rational numbers is a rational number. c. The product of two irrational numbers is an irrational number. d. The product of a rational number and an irrational number is an irra

Question Regarding Einstein Summation Convention

Hello. I had a quick question relating to a confusion I had about the Einstein summation convention: Is writing: h_{ab}u^{a}u^{b} The same as writing: h_{ab}u^{b}u^{a}? Where "_" is a subscript, and ^ indicates superscript. Thanks!

Solutions to Two Systems of Nonlinear Equations

Solve the following system of nonlinear equations for the unknown angles alpha, Beta, and lambda, where 0 <= alpha <= 2*pi, 0 <= Beta <= 2*pi, and 0 <= lambda < pi. 2 sin(alpha) - cos(Beta) + 3tan(lambda) = 3 4 sin(alpha) + 2cos(Beta) - 2tan(lambda) = 2 6 sin(alpha) - 3cos(Beta) + tan(lambda) = 9 Solve the following sys

Example Linear Algebra Problems

Let 0 denote a 2 x 2 matrix, each of whose entries is 0. a) Is there a 2 x 2 matrix A such that: A does not equal 0 A*A=0? b) Is there a 2 x 2 matrix A such that: A does not equal 0 A*A=A?

Construct a Truth Symbol

In exercises 59-62, let p: Tanisha owns a convertible. q: Joan owns a Volvo. Translate each statement into symbols. Then construct a truth table for each and indicate under what conditions the compound statement is true. #62. Tanisha does not own a convertible or Joan does not own a Volvo.

Correction and Comment on Linear Algebra Problems

With referrence to the last solutions u posted to me,may you please check no 1 I have a different solution from my workings.I not sure which one is correct,but I think mine might be.If so may u please edit the document because I failed to do so.I will use this over and over.

Eigen function and eigenvalues

I have attached a problem involving eigenfunctions and eigenvalues. I would sure appreciate a complete explanation with work. Consider a linear operator ... with two sets of eigenfunctions and eigenvalues: ... with ... Now consider a linear combination of eigenfunctions, ... . Show whether or not ... is an eigenfuncti

Solve the system of simultaneous equations.

Solve each system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with y arbitrary. 1. 3x+2y=5 6x+4y=8 2. 3x+5y= -2 9x+15y= -6

Matrices Eigenvalues, eigenspace, and eigenbasis.

Matrices Eigenvalues, eigenspace, and eigenbasis. See attached file. For each of the matrices, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis, if you can. [■(7&8@0&9)] [■(1&1@1&1)] [■(6&3@2&7)] [■(0&-1@1&2)] [■(1&1&0@0&2&2@0&0&3)] [■(1&1&0@0&1&1@0&0&1)]

Solution to Matrix

Solution to matrix: real-values fundamental set of solutions, equilibrium, the type and stability characteristics of the equilibrium point, 1st order linear ODE. Need complete explanation and work. Thanks. Please see the attached file.

Coin Combination Probability

Suppose you have 50 American coins totaling exactly $1. This includes at least one quarter, and may contain nickels, dimes, and/or pennies. If you drop a coin at random, what is the probability that it is a penny?

Tetrahedral and Octahedral Groups

Let G=O be the group of rotations of a cube. Two regular tetrahedra can be inscribed in this cube, each using half of the vertices. Let H be the subgroup carrying one of the two inscribed tetrahedra to itself. If T is the tetrahedral group, show that H=T.

proofs about determinants

Prove: 1) if A, B are square and AB=I_n, then det B=1/det A 2) if A, B, S are square and B=(S^-1)AS, then det B=det A 3) If A is square and (A^T)A=I_n, then det A= +/- 1 4) if A is n x n and A^T= -A, then det A= (-1)^n det A. In particular, if n is odd, then det A=0

Polynomial Linear Operator

Let T: P_3 --> P_3 be defined bu (Tp)(t) = p(t + 1). a) Show that T is a linear operator. b) Find the nullspace and range of T. c) Let Beta = (1, 1+t, 1+t+t^2, 1+t+t^2+t^3). Show that Beta is a basis for P_3. d) Find M(T,Beta, Beta). e) Find the eigenvalues and eigenvectors of T and give the characteristic and minimal

Prove that f(c) is the minimum value of f in I.

3. Let I be an interval in R and assume f : I ? R is twice differentiable at all points of I. Suppose that a ? I, f?(a) = 0 and f?(x) ?0 for all x ? I, x ? a. Prove: If f??(a) > 0, then f(c) is the minimum value of f in I; that is f(c) ? f(x) for all x ? I.

Solve the system of linear equations by substitution

Alice's Restaurant has a total of 205 seats. The number of seats in the non-smoking section is 73 more than twice the number of seats in the smoking section. How many seats are in each section? Please show work.

Manufacturer produces three types of radios

First equation: A company makes three products X, Y and Z. Each product requires processing by three machines A, B and C. The time required to produce one unit of each product is shown below. PRODUCT MACHINE A MACHINE B MACHINE C X 1 2 2 Y 2 8 3 Z 2 1 4 The machines are available for 200, 525 and 350

MAT 101

This assignments consists of two parts in the first part I want you to create a linear equation and in the second part I want you to create a system of equations. Systems of Equations (Part I) Think of a mathematical function that represents something from your own life. For example, suppose the number of beers you drink d

Linear Spaces and Isomorphism

See the attached file. 1. Find a basis B of R^n such that the B-matrix B of the given linear transformation T is diagonal. Reflection T about the line in R^n spanned by [2;3 2. Which of the subsets of P_2 given below are subspaces of P_2? Find the basis for those that are subspaces. i) {p(t):p(0)=2} ii) {p(t):p(2)=0} i

Cholesky Decomposition and Schur Compliment

Please provide detailed proof. 7. Let k and l be positive integers, and set n:= k + l. Suppose A <- C^nxn has the decomposition A = | B C^H | | C D |, where B <- C^kxk, B <- C^lxk, and D <- C^lxl. (a) If A is HPD, prove that B, D and E :: D - (C B^-1 C^H) are HPD. E is called the Schur compliment of B in