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Linear Transformation

Graphical Minimization Method

The ABC small-scale industry has production facilities for two different products. Each of the products requires three different operations: grinding, assembling, and testing. Product 1 requires 30, 40, and 20 minutes to grind, assemble, and test, respectively; product 2 requires 15, 80, and 90 minutes for grinding, assembling,

Estimating the Price at Which Revenue Is Maximized

Demand for pools. Tropical Pools sells an above ground model for p dollars each. The monthly revenue for this model is given by the formula R(p)=0.08p2 +300p. Revenue is the product of the price p and the demand (quantity sold). a) Factor out the price on the right-hand side of the formula. b) Write a formula D(p) for th

Integer program model to determine which investments

Investment Cost Expected Return Condition 1 $200,000.00 $25,000.00 only if 3 2 $1,000,000.00 $100,000.00 not if 1 3 $750,000.00 $150,000.00 none 4 $1,000,000.00 $270,000.00 only if 1 and 3 5 $500,000.00 $300,000.00 not if 3 6 $500,000.00 $100,000.00 none Formulate an integer program model using S

Understanding Kernel

Please help me to understand the attached homework problem on finding the kernel. Find the Kernel Consider the groups of nonzero complex numbers, *, and positive real number +, both with multiplication. Let f: *  + be defined by f(z)= |z|. Prove that f is a homomorphism and find

Symmetric and Galois group

Let S_3 be the symmetric group on the set {1,2,3}. Show that S_3 is solvable and that the Galois group Gal(P/Q) of the polynomial P=Y^3+pY+q over Q is a subgroup of S_3. (See attached)

Category Theory - Morphisms, Uniqueness & Equivalence

If f:A-->B is an equivalence in a category C and g:B-->A is a morphism such that gf=1_A (the identity on A) and fg=1_B, show that g is unique. Prove that any two universal (initial) objects in a category C are equivalent. Prove that any two couniversal (terminal) objects in a category C are equivalent. In the category o

Schur's Lemma Implies Functions

I have some trouble understanding the solution to the attached problem (solution included). Could you please provide some clarification of the solution. I have indicated what my points of concern are. Show that if M1 and M2 are irreducible R modules, then any nonzero R-module homomorphism from M1 to M2 is an isomorphism. De

Universal pair proofs

Definition: A pair (U, epsilon) is universal for a group G, with respect to abelian homomorphic images, if U is an abelian group, epsilon:G --> U, an epimorphism, such that for any other abelian group A and surjection f:G --> A, there exists a unique g:U --> A such that f=g composed with epsilon. In this case, we say f can be

Rigorous Subgroup Proofs

I need to see rigorous proofs of these propositions. 1.) Suppose [G:H] is finite. Show that there is a normal subgroup K of G with K, a subgroup of H, such that [G:K] is finite. 2.) Suppose H is a subgroup of S_n but H is not a subgroup of A_n. Show that [H:A_n intersect H]=2. 3.) Prove that if H,K are

Module homomorphism proof

Let φ: Z "circle +" Z → Z "circle +" Z be a module homomorphism (of Z-modules). Show that if φ is surjective, it must be injective. Give an example to show that the converse is false ─ a difference between free Z-modules and vector spaces. (You may, of course, think of φ as a 2 x 2 matrix with integer

Linear transformation

Please show work. Consider R^p with the standard inner product....Prove that the orthogonal projection...is a linear transformation...Is the transformation one-to-one (injective)? Is the transformation onto (surjective)? Justify your answers.

Consider the Linear Transformation T: Complex Numbers

Consider the linear transformation T : complex numbers^n -> complex numbers^n given by T( z1, z2, ... , zn ) = ( a1z1, a2z2, ... , anzn). What is the dimension of the subspace spanned by the eigenvectors of T? Exhibit a basis for this space, and give the eigenvalues.

Transformation Matrix in Vector Calculus

Section 6.1 The Geometry of Maps from R^2 to R^2. Please see the attached pdf. file. Let D* be the parallelogram with vertices...Find a T such that D is the image set of D* under T.

Automorphism proof

Please show all work. Thanks See attached Task: Prove the following theorem: is an automorphism of under componentwise addition. Additional Notes on the Task: Be careful not to confuse the homomorphism's action as a function with the group operation's action as componentwise addition. is componentwise ad

Modelling Problem Using Excel

A company that makes bikes wants to maximize profit over the next five months. Materials for each bike costs $600. Both humans and machines are needed to produce each bike. Each human can work on up to 100 bikes per month. Each machine can work up to 200 bikes per month. TetraCon, Inc. has 4 humans and no m

Minimizing travel relocation

A quarry consists of 8 quarry pits along a 4000 yard north-south section of land. Each pit is 500 yards away from the other. Rock must be removed from some quarry pits and relocated to others quarry pits as listed below. Quarry 1: 0 to 500 yards - 7,000 tons of rock are needed Quarry 2: 500 to 1000 yards - 3,000 to

Closest Point to the Origin

Find the closest point of the surface: xy + xz + yz = 1 to the origin. and x^2 + y^2 - Z^2 = 1 to the origin.

Convexity and extreme values

Check that the function: f(x1, x2, x3) = (x1)^2 + (x2)^2 + (x3)^2 - x1 - x2 - x3 is convex. Find the extreme values of f under the conditions: (x1)^2 + (x2)^2 = 4, -1 <= x3 <= 1. (x3 goes from -1 to 1)

Converting temperature between Celsius and Fahrenheit

Temperature can be given in two ways: Celsius and Fahrenheit. Using the Library, web resources, and other course materials, find the equations that show the relationship between Celsius and Fahrenheit. Give both equations (one that solves for F and one that solves for C) and explain why each can be beneficial to use. The re

Maximizing the Volume of a Rectangular Box

You are planning to make an open rectangular box from a 12-by 14-cm piece of cardboard by cutting congruent squares from the corners and folding up the sides. a) What are the dimensions of the box of larges volume you can make this way? b) What is its volume?

Transformation matrices

See attached file. Consider the relationship between the matrices A and B, explore the relationship between A and B and matrices representing other linear transformations derived from F and G. Explore 2 examples for example &#61537;F+&#61538;G for various &#61537; and &#61538;, or the composite function F&#61616;G/ Check th

Linear transformations

See attached file. 1. For each of the following functions decide whether or not the function is a linear transformation.

Linear Transformation: Minimization

Boise, Idaho is about 300 miles inland from the nearest point on the Pacific coast; San Diego is about 1000 miles south of that point down the coast. Assuming the coast is a straight line going north-south, C is the point along the coast directly west of Boise. It costs 2 cents per mile to transport a ton of potatoes by truck an

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

Minimal polynomial and stabilizer

Any explanations of the attached definitions is greatly appreciated. Thank you. The definition of the minimal polynomial for the linear transformation.... the relationship between the cosets in G of the stabilizer....

Fields and homomorphisms

Not sure why, but I am having trouble with this one. I'd really appreciate it if someone is willing to help here. Please see the attached file. Thank you