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# Linear Transformation

### Maximizing the Revenue for an Airline Company

An airline has a new airplane that will be fitted out for a combination of first and second class passengers. A first class seat will cost \$120 on a certain one-way trip and a second class seat \$80. The seating capacity of the plane is 200 second class seats. A first class seat takes 1.2 times the floor area of a second-class se

### Linear Programming Model for Maximizing Profit of a Production Schedule

Ms. Olsen, a coffee processor, markets three blends of coffee. They are Brand X, Minim and Taster's Reject. Ms. Olsen uses two types of coffee beans, Columbian and Mexican, in her coffee. The following chart lists the compositions of the blends. Blend Columbian Beans Mexican Beans Brand X 80% 20% Minim 50

### Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint

Please see the attached file for the fully formatted problems. Find a 5x5 matrix M>0 such that if and x(t)= then Can we use this definition to find the adjoint of T (T is given at the end)? This part is the additional information to solve the question above; Let V=L2[-1,1] be the Hilbert space of func

### Transformation of State Variables and State-Space Representations

Consider the system . It can be shown that there is other state equation possible for this system. Let us define a new set of state-variables z1 , z2 , z3 by the transformation: Obtain the state-space representation of the system in terms of the new variables. Please see the attached file for the fully formatted problem

### Graph theory: Representing the Linear Transformations

Please see attached file for full problem description. Let C0 ={SUM (i = 1 to p) &#949;ivi&#9474;&#949;i is an element of F2, vi is an element of V(G)} be the vector space of 0-chains and Let C1 ={SUM (i = 1 to q) &#949;iei&#9474;&#949;i is an element of F2, ei is an element of E(G)} be the vector space of 1-chains Re

### Linear Spaces, Mappings and Dimensional Spaces

1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X. 2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V). Show that the dimension of the null space of ST is less than or equal to the sum of the dimensions of the null spaces of S and T. 3)

### Linear Mappings, Differentiation and Linear Spaces

Please help with the following problems. Provide step by step calculations for each. 1) Show that this mapping is linear: T: P5 -> P8 defined as Tp(t)=p(t+1)-p(t)+integral(t-1 to t) s^2 p(s) ds 2) Prove the following is true, or give a counterexample: If l is a nonzero scalar linear function on linear space X (which may

### Solve: Inventory Management

1. Cooper Automotive Products manufactures components used in the automotive industry. The company purchases parts for use in its manufacturing operation from a variety of different suppliers. One supplier provides a part where the assumptions of the EOQ model are realistic. The annual demand is 5000 units, the ordering cost i

### 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

### 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 : 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

### 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

### 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

### 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 .

### 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

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

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

### 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)

### 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

### Linear Transformations, Change of Basis and Conjugation

Let V = Q3 and let ' be the linear transformation from V to itself: '(x, y, z) = (9x + 4y + 5z,&#8722;4x &#8722; 3z,&#8722;6x &#8722; 4y &#8722; 2z), x, y, z E Q With respect to the standard basis B find the matrix representing this linear transformation. Take the basis E = {(2,&#8722;1,&#8722;2), (1, 0,&#8722;1), (3,&#8722;2

### Linear Transformations : Basis, Kernel, Image, Onto, One-to-one and Matrices

1) Define a linear transformation.... a) Find a basis for Ker T. b) Find a basis for Im T. c) Is T an onto map? d) Is T a one-to-one map? 2) Define a linear transformation... a) Find the matrix for T with respect to the standard basis. b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t

### Order of Elements, Factor Groups and Homomorphisms

Help with this linear transformation problem. Please help with the following problem. Provide step by step calculations. A. What is the order of the element 14 + <8> in the factor group Z24 / <8>? I know that if I let G = Z24 and H = <8> , then H = <8> = {0,8,16}. So Z24/<8> = {0+H,1+H,2+H,3+H,4+H,5+H,6+H,7+H}.

### Job Shop Problem

Suppose you have N jobs that have to be processed on a single machine. For i = 1, 2, . . . ,N, job i requires pi units of time on the machine, and has weight wi. The objective is to schedule these jobs so as to minimize the sum of the weighted completion time of all the jobs, where the completion time of job i is the time at w

### Using Excel Solver : Maximizing Interest and Cash Flow

At the start of the year a company wants to invest excess cash in one-month, three-month and six-month CD's. The company is somewhat conservative and wants to make sure it has a safety margin of cash on hand each month. (left over from previous month/ available at the outset, plus principal and interest from CD's that have becom

### Topology : Homomorphism Formatted

Please see the attached file for the fully formatted problems. B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant. (b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism. (c) (i) Recall that Fr E, the fron

### Scatterplots for Tree Growth

Tree growth scatterplot. Age 5, 14, 29, 16, 16, 26, 6, 25, 7, 18 Diameter 8, 23, 34, 24, 24, 10, 30, 14, 13 Summary points for first middle and last group based on median-median.(show cords) Equation of the line passing through the summary points. Calculate distance from line through the outer summary points to the middle p

### Kernel and Homomorphism

Here's my problem: If A and B are subsets of a group G, define AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel. (i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed accordingly; do not assume that G is abelian.) (i

### Homomorphisms

Please assist me with the attached homomorphism questions. Thanks! Example: ? Let f: G -->H be a group homomorphism with kernel K = Ker(f), show that f is one to one if and only if K = ...

### Find a self-complementary graph with five vertices.

Let G be a graph. Then G = (V, E), where V and E are the vertex set and edge set, respectively, of G. The complement of G, which we will refer to as "G bar," is the graph (V, E bar), where V is the vertex set of G bar (i.e., the vertex set of G bar is identical to the vertex set of G) and E bar is the edge set of G bar. The e

### Prove that graphs that are isomorphic have the same number of vertices and the same number of edges, and that the degree of a vertex of a graph is equal to the degree of the image of that vertex under a graph isomorphism. Also, give an example of a pair of non-isomorphic graphs that have the same number of vertices and the same number of edges.

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphi