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# Linear Transformation

### Simulation-Assigning Random Numbers

A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historic

### What are the total monthly transportation costs for the optimal solution?

A logistics specialist for Wiethoff Inc. must distribute cases of parts from 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are: Destination Assembly Plant 1 2 3 Supply Source A 5 9 16 200 Factory B 1 2 6 400 C 2 8 7 200 Demand 120 620 60 What are the total monthly tr

### Transportation problem: Cost minimization

Three quarries produce lime which is then shipped to five warehouses Quarry Tons Lime 1 200 2 100 3 150 Transport Costs (per ton) Quarry 1 2 3 4 5 1 \$5 \$1 \$6 \$3 \$1 2 \$2 \$3 \$4 \$5 \$4 3 \$4 \$2 \$3 \$2 \$3 Demand 80 90 100 70 60 (tons) Minimize t

### Range of Linear Operators

Show that the range of the linear operator defined by the equations w1 = x1 - 2x2 + x3 w2 = 5x1 - x2 + 3x3 w3 = 4x1 + x2 + 2x3 is not all of R3, and find a vector that is not in the range.

### Splitting Fields

Let E be an extension of F and f(x) be in F[x]. Also, let &#934; be an automorphism of E leaving every element of F fixed. Prove that &#934; must take a root of f(x) lying in E into a root of f(x) in E. See attached file for full problem description.

### Excel Solver - maximize the annual passenger-carrying capability.

An airline owns an aging fleet of jet airplanes. It is considering a major purchase of up to 17 new model 7a7 and 7b7 jets. The decision must take into account numerous cost and capability factors, including the following: (1) The airline can finance up to \$400 million in purchases; (2) each 7a7 jet will cost \$35 million and eac

### Group Homomorphism and Abelian Groups

Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi). Proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel. Please show how to do the reverse (<=) and show that phi is

### Homomorphism Subgroup Proofs

Let G be a group and let H be a normal subgroup of G. Let m be the index of H in G (that is, the number of cosets of H). Prove that for any a we have am H. (b) Give an example of group G, a subgroup H of index in, and an element a G such that am is not in H. (Of course, your subgroup H had better not be normal.) (4) (a) Suppos

### Linear Transformations, Rotations, Submodules and Subspaces

(7) Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by pi-radians. Show that every subspace of V is an F[X]-submodule for this T. Here F[X] is a polynomial domain where the coefficient ring is a field F.

### Linear Transformations : The composition of any two reflections

Matrix Theory - Homework 7 Prove the following in several stages: The composition of any two reflections,whose lines of reflection are orthogonal,is a half-turn. We will work in the vector space &#8477; .... First Stage Refer to the diagram provided.The line l makes some angle &#57534;with the x-axis. Let us suppose that

### Automorphisms and Conjugation

Show that if H is any group then there is a group G that contains H as a normal subgroup with the property that for every automorphism f of H there is an element g of G such that the conjugation by g when restricted to H is the given automorphism f, i.e every automorphism of H obtained as an inner automorphism of G restricted to

### Homomorphisms and Surjections

Let f:G->H be a group homomorphism. Prove or disprove the following statement. 1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order. 2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G. F

### Homomorphisms and Mapping Prime Elements

Let G and H be two finite cyclic groups of relatively prime orders. Determine the number of homomorphisms from G to H.

### Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups

Problem 1. Let a,b be elements of a group G Show a) the conjugate of the product of a and b is the product of the conjugate of a and the conjugate of b b) show that the conjugate of a^-1 is the inverse of the conjugate of a c)let N=(S) for some subset S of G. Prove that the N is a normal subgroup of G if gSg^-1<=N for

### Maximizing Revenue: Example

A cable TV company has 12,000 customers and charges \$16 per month for basic service. The managers believe they will lose 400 customers for every dollar they raise the monthly charge. Determine the monthly charge that will maximize their revenue.

### Composition of Functions and Isomorphisms

I can't prove the following statements about functions f:A->B and g:B->C 1. If gof is one-to-one then so is f. 2. If gof is onto then so is g. Furthermore I don't know how to show that f: A->B is an isomorphism of sets if and only if there is a function g: B->A such that gof=1A and fog=1B. Here fog and gof are compossiti

### Maximizing Profit Problem

Suppose that a laundry service determines that to attract x customers per day, its price of service must be 3.20 - 0.02x dollars. If their total cost to serve x customers is 0.05x^2 + 1.10x + 120 dollars, how many customers should be served to maximize their profit? (Hint: Use the price to help get the revenue)

### Linear Transformations and Subspaces

B1) This question concerns the following two subsets of : (a) Show that , and find a vector in that does not belong to T. [3] (b) Show that T is a subspace of . [4] (c) Show that S is a basis for T, and write down the dimension of T. [7] (d) Find an orthogonal basis for T that contains the vector .

### Maximizing the Sustainable Yield

3.17) A lake has a carrying capacity of 10,000 fish. At the current level of fishing, 2000 fish per year are taken and the fish population seems to hold fairly steady at about 4000. If you wanted to maximize the sustainable yield, what would you suggest in terms of population size and yield?

### Urgent

Please help. I only need answers with brief explanations. No need of detailed working. (See attached file for full problem description) --- State whether the following are true or false with reasons: 1. If a in S6, then an =1 for some n greater than or equal to 1. 2. If axa-1=bxb-1, then a=b 3. The function e

### Cyclic Vector

Suppose that T is a linear operator on a two dimensional vector space V, and that T is not multiple of the identity transformation. Show that T has a cyclic vector (i.e., there exists v V such that {v, Tv } is a base for V). Please see the attached file for the fully formatted problems.

### Isomorphism proof

Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism. Recall The the function f is an isomorphism if and only if f is onto and Kernel ={ 0}. Please explain step by step with reasons in every step.

### (N)T is a normal subgroup of G.

Modern Algebra Group Theory (LXVI) Automorphism of a Group Let G be a g

### Automorphism of a group

Modern Algebra Group Theory (LXI) Automorphism of a Group Is the mapping given below an automorphism of the group ? G group of integers under addition, T:x --> -x

### Isomorphisms and path classes

(See attached file for full problem description with proper symbols) --- a) Under what conditions will two path classes, and , from to , give rise to the same isomorphism of onto ? b) Let be an arcwise-connected space. Under what conditions is the following true: For any two points , all path classes from

### Let G be any group, g a fixed element in G. Define phi:G--> G by phi(x) = gxg^-1. Prove that phi is an isomorphism of G onto G.

Group theory Modern Algebra Group Theory (LV)

### Closed Graph, Banach Spaces and Continuous Linear Transformations

Let and be Banach spaces and let . (Note: is a set of all continuous linear transformations ). Show that there is a constant such that for iff and gra A is closed. ---

### M/M/s Queue, Stationary Distribution and Long-Run Proportion

Consider the M/M/s queue, with arrival rate... 5. Consider the M/M/s queue, with arrival rate Λ >0 and service rate μ >0. (a) Find the condition involving Λ, μ and s that is necessary for there to exist a stationary distribution. Why does this condition makes sense? (b) Find the long-run proportion of time that there ar

### Isomorphism

We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5. Lemma: Let H be a normal subgroup of a finite group G, and let x be an element of G. If o(x) and [G:H] are relatively prime, then x is in H. Theorem: Any 60 element group having 24 el

### Linear Mapping, Linear Space, Differentiability and Continuity

In each of Exercises 40 through 46 following, a linear space V is given and a mapping T : V&#8594;V is defined as indicated. In each case determine whether T is a linear mapping. If T is linear, determine the kernel (or null space) and range, and compute the dimension of each of these subspaces wherever they are finite-dimension