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Linear Algebra

Boolean product of matrices, One to one function

** Please see the attachment for complete questions (1 and 2) ** 1. Find the Boolean Product of matrices A and B. 2. Given matrix A, compute: (a) A^-1 (A-inverse) (b) (A^-1)^3 3. Solve the following systems of equations. x1 + x2 = 0 -x1 + x2 + x3 = -1 -1x2 + x3 = 2 4. (a) Define the function f: R -->

Show divisibility.

1. Emperor's Banquet You have been invited to the emperor's banquet. The emperor is a rather strange host. Instead of sitting with his guests at a large round dining table, he walks around the table pouring oatmeal on the head of every other person. He continues this process, pouring oats on the head of everyone who has n

System of Equations, Index Numbers

Solve the system of equations: 3x + y = -7 x - y = -5 Solve the system of equations: 3x + 2y = 18 y = 3x Which of the following is true of a base period for an index number? The numerator spears It appears in the denominator It must have occurred after the year 1980 It cannot be less than 100

Algebra: vector space

Determine which of the following subsets of M22 form subspaces. a) The subset having diagonal elements nonzero. b) The subset of matrices whose (1,2) element is 0. c)The subset of matrices of the form ( a b ) (-b c ) d)The subset of 2x2 matrices whose determinants

Is a weak contraction a contraction?

Let M be a subset of a metric space (X,d). A function f: M -----> M is a weak contraction provided that for any x not equal to y in M, d ( f(x), f (y) ) < d ( x, y). a) Is a weak contraction a contraction? ( Proof or counterexample) b) If M is compact. Is a weak contraction a contraction? ( Proof or counterexample)

preserve distance for translation map

In R2, let C be any point. Let T_c : R2 --> R2 be the translation map T_c(P) = P + C. (a.) Show T_c is a bijection by showing it is one to one and onto. (b.) Show T_c is a bijection by finding its inverse. (c.) Show T_c is distance preserving (using vectors).

Proving that a closed unit is not compact

Prove that the closed unit ball of the normed space (C[0,1], | |infinity) is not compact. As usual C[0,1] stands for the space of all continuous functions f: [0,1] -> R, and | |infinity is the uniform norm on that space. Please refer to the attachment for appropriate notations in the question.

Proof for Non-Empty Subsets

Please help with the following problem. Use the following steps to prove that every non-empty open subset of R is a union of at most countably many disjoint open intervals. Suppose that G is a non-empty open subset of R. 1. For each a <- G let Ia be the union of all those open intervals I which contain a and are containe

Accurately Sketching a Phase Portrait

Classify the fixed point at the origin and accurately sketch a phase portrait for each of the following systems (i) dx/dt=x-2y dy/dt=3x-4y (ii) dx/dt=3x+2y dy/dt=-2x-2y (iii) dx/dt=-x-5y dy/dt=x+3y (iv) dx/dt=x+y dy/dt=-2x-y (v) dx/dt=2x dy/dt=-

Assessing the Field of Quotients

Let R = Z[x]. Let f : R -----> C be defined by evaluation of polynomials at sqrt (-5), which we know to be a homomorphism. a. What is the kernel of f? why? b. the image of f is a subring of C. describe it and its field quotients. No proof.

Find all the eigenvalues of a matrix

Please see attachment for full problem description and hint. The solution of the vector-valued differential equation dX/dt = AX(t) with X(0) = [a1 a2 a3] is given by: X(t) = Exp(tA)X(0) = e^tA(X(0)) Explain how you would compute e^

Linear Algebra - Singular Matrix

Let a,b,c,d,e,f,g,h be reals. Prove that the following matrix A is singular. Matrix A: 0 a 0 0 0 b 0 c 0 0 0 d 0 e 0 0 0 f 0 g 0 0 0 h 0

Determining system reliability

A system consists of three subsystems in parallel (assume operating redundancy). The individual subsystem reliabilities are as follows: Subsystem A = 0.98 Subsystem B = 0.85 Subsystem C = 0.88 Determine the overall system reliability

Systems of Linear Equations - Graphs

Answer the question and solve the problems below. Make sure you show all your work. 1. Determine whether the lines will be perpendicular when graphed. 3x - 2y = 6 2x + 3y = 6 2. Alice's Restaurant has a total of 205 seats. The number of seats in the non-smoking section is 73 more than twice the num

Linear Functions #2

Think of a mathematical function that represents something from your own life. For example, suppose the number of beers you drink depends on the number of football games you watch. If you drink five beers during every football game, the function would be: Number of Beers (B) = 5 times Number of Football Games (F), or B = 5F

Determining the Number of Calculators Produced

The cost C, in dollars, to produce calculators is given by the function C(x)=57x+4500, where x is the number of calculators produced. How many calculators can be produced if the cost is limited to $61,500?

MAPE and MAD Linear Regression Model

Senior Management at Esil University believes that decreases in the number of undergraduate applications that they have experienced are directly related to tuition increases. They have collected the following enrollment and tuition fees data for the past decade: Year: Undergraduate Applications: Annual Tuition

Augmented Matrices and Row Operations

Given the following matrices/vectors: 2 4 1 (-1) 4 a:=0 b:=(-2) A:= 3 1 0 1 6 1 2 1 linear systems: find x,y such that Ax=b and Ay=a (note x, y with vector character over each).

Matrices, Eigenvalues and Eigenvectors

Let M=[(4 -1 -9 -5), (3 -2 -5 -3), (3 0 -7 -3), (0 -1 1 -1)] i) Show that the column vector [7,5,3,2]^T is an eigenvector for M. ii) Find the eigenvalues for M. iii) Determine whether or not M is diagonalisable over R, justifying your assertion and showing any necessary calculations in full.

Problems with normal subgroups

Let N_1, ..., N_k be normal subgroups of a finite group G. If G = N_1N_2 ...N_k (the set of all elements of the form a_1a_2...a_k with a_ j in N_ j) and |G| = |N_1| . |N_2| . |N_3| ... |N_k|, prove that G = N_1 x N_2 x N_3 x ... x N_k.

system of linear equations..

Total number of cherries on a plate, in a cup, and in a bowl is 780. The plate contains 145 cherries. The number of cherries in the bowl is 4 times the total number of cherries on the plate and in the cup. How many cherries in the cup?

Systems and equations

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.,

Linear Algebra

P 3 is a vector space of polynomials in x of degree three or less and Dx(p(x)) = the derivative of p(x) is a transformation from P 3 to P 2. a. the nullity of Dx is two. b. The polynomial 2x + 1 is in the kernel of Dx. c. The polynomial 2x + 1 is in the range of Dx. d. The kernel of Dx is all those polynomials in P 3 with

Linear combination problem

If T: U â?' V is any linear transformation from U to V and B = {u 1, u 2, ..., u n} is a basis for U, then set T(B) = {T(u 1), T(u 2), ... T(u n)} a. spans V b. spans U c. is a basis for V d. is linearly independent e. spans the range of T

Eigenvalues of the Transition Matrix

A discrete dynamical system model for the population of cheetahs and gazelles in Namibia is given by the following pair of equations: .3Ck + .4 Gk = Ck+1 -pCk + 1.3 Gk = Gk+1 where Ck measures the number of cheetahs present in a certain Namibian game reserve at time K, Gk gives the number of gazelles (measured in tens), a

Linear Algebra: Systems of Equations

Please show all work for the attached questions. 8. A cellular phone company offers a contract for which the cost c, in dollars, of t minutesof telephoning is given by C= 0.259t-300)+79.95, where it is assumed that >300 minutes. What time will keep cost between $129.95 and $167.20? - For the cost