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# Linear Algebra

### 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

### Proving a function is irreducible

Let K be a field and let f(x) in K[X] be a polynomial whose degree is either 2 or 3. Show that f(x) is irreducible if and only if it has no roots.

### 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

### Create a Real World Linear System

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

### 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

### Systems of Equations Homework Help

Solve the following systems of equations using the method outlined in week 7 (Part I) of the notes. Your procedure should be in matrix form as was done in Example 3. (Read your notes carefully before attempting the problem and simply follow the examples) x1 + x2 = 2 -x1 + x2 + x3 = 0 -1x2 + x3 = 1

### Properties of Linear Operators

In each of parts (a) and (b), an operation * is defined over the set of natural numbers. For each operation, determine these four things. See Attachment. Please provide detailed explanation showing all steps and reasoning as well as formal notation for the Proof. Please post response as a MS Word or PDF file. Thank

### Car rental: Find and solve a system of equations

A major city car rental agency has a total of 2200 cars rented out of 3 locations: Metro Airport (m), Downtown Airport (d), and at smaller City Airport (c). Weekly rental and return patterns are in a table. How many cars should be at each location so that the same number of cars will be there at the end of the week?

### Conjugate Subgroups and Conjugacy Classes

1.) If H and K are conjugate subgroups, show that N(H) and N(K) are also conjugate. 2.) If G is a finite group with only two conjugacy classes, show that |G| = 2. NOTATION: "N(X)" is the normalizer of group X.

### Householder Reflector - Proof in Numerical Linear Algebra

1. Determine the eigenvalues, determinant, and singular values of a Householder reflector. Give algebraic proofs for your conclusions. 2. Suppose Q E C^n, llqll2 = 1 Set P = I - qq^H. (a) Find R(P) (b) Firrd l/(P). (c) Find the eigenvalues of P. Prove your clairns. 3. Let A E C^(m*n)}, m (greater or equal to) n, with ran

### Analyze morphisms.

Show that a contravariant functor takes morphisms that admit left inverses to morphisms that admit right inverses and vice versa.

### Hilbert space, operator, sequence

Please see the attached file! Thanks!! A is bounded means ||Ah|| less than or equal to C||h|| for all h A* is the adjoint of A, i.e. <Ah|k>=<h|A*k> A is greater or equal to 0 means <Ah|h> is greater or equal to 0 for all h

### Problems Involving Linear Equations and Functions

See the attached file. Race was 46.2 sec in 1920. In 1940 was 46.0 sec. Let R (t) = the record in the race and t= the number of years since 1920. I need help finding R (t) = what by rounding to nearest hundredth Predicted record for 2003 is ____ sec (round to nearest hundredth) Predicted record for 2006______sec (rou

### Assess the sets.

Given the following sets, select the statement below that is true. A = {l, a, t, e, r}, B = {l, a, t, e}, C = {t, a, l, e}, D = {e, a, t} B â?â?? C and C â?â?  A C â?â?  B and D â?â?? B D â?â?? A and A â?â?? D B â?â?? A and C â?â?? D D â?â?  A and A â?â?? C

### Prove the Perturbation Estimate

1. Let A <- C^nxn be invertible and suppose b <- (C^n)_*. Suppose x <- C^n satisfies Ax = b. Let the perturbations dx, db <- C^n satisfy A dx = db, so that A (x+ dx) = b + db. (a) Prove the error (or perturbation) estimate (1 / cond(A)) ( lldbll / llbll ) <= ( lldxll / llxll ) <= cond(A) ( lldbll / llbll ) . (b) Show t

### Solutions of a System of Nonlinear Equations

Give an example of a nonlinear system of two equations in two unknowns having: (a) exactly 6 solutions Verify your work and give points of intersection. (b) exactly 7 solutions Verify your work and give points of intersection. Based on your work, how would you devise a method for finding such a system having exa

### Proof in Linear Algebra with Matrices

6. Suppose A = [a,b: b,a] where a and b area real numbers. Show that the subordinate matrix norms satsify ||A||1 = ||A||2 = ||A||_infinity.

### Finding a 2 x 3 system of equations

Consider the set of points (x, y, z) defined by the set of equations below: { x = 1 - t } { y = 2t } { z = 1 + t } where t is a real number parameter. Find a 2 x 3 system of linear equations having this set of points as its solution set. Please show all steps in your work and explain your answer in detail.

### Systems of Linear Equations using personal life experiences

For this SLP I want you to create a system of linear equations from your own life, it can be an extension of your module 2 SLP or something new entirely. Keep in mind that a system of linear equations will consist of two equations using the same variables and the variables will represent the same thing for both equations, i.e.,

### Systems of Linear Equations

Module 3 - Case Systems of Linear Equations Answer the question and solve the problems below. Make sure you show all your work so you can get partial credit even if you get the final answer wrong. 1. Determine whether the lines will be perpendicular when graphed. 3x - 2y = 6 2x + 3y = 6

### Proof in Numerical Linear Algebra Explained

Suppose that C [an element of] C^nxn is invertible and the set S = {w_1,...w_k} C [a subset of] C^n is linearly independent. Prove that CS := {Cw_1,...Cw_k} C [a subset of] C^n is linearly independent. See the attached file.

### Quantitative Methods Linear Equation a-f

5) A small manufacturer produces special order DVDs for individuals and businesses. The materials and labor costs for each DVD is \$0.25. The fixed costs of production are \$525. The manufacturer charges a client \$5.00 for each DVD. Suppose x denotes the number of DVDs produced and sold. Use the information above to find the follo

### function and system of linear equations

Find the coordinates of the x-intercept. 2x + y = -6 Find the y-intercept. -2x - y = -10 Find the coordinates of the x-intercept. -x + 5y = -15 Find the domain and range of the relation, and state whether or not the relation is a function. {(1, 4), (1, 5), (1, 6), (1, 7)} Determine whether the system is cons

### Left and Right Eigenvectors with Distinct Eigenvalues

Let A belonging to C^nxn be a skew-hermitian. a) Prove directly that the eigenvalues of A are purely imaginary. b) Prove that if x and y are eigenvalues associated to distinct eigenvalues, then they are orthogonal, i.e. x^H*y = 0