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Linear Transformation

A linear transformation or linear map is a function between two modules that preserves the operations of module addition and scalar multiplication. As a result, it always maps linear subspaces to linear subspaces. For example it maps straight lines to straight lines or a single point. The expression linear operator is often used to refer to a linear map from a vector space to itself. A linear map is a homomorphism of modules. It is a morphism in the category of modules over a given ring.

For example, let V and W be vector spaces over the same field K. A function f: V→W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, then the following two conditions are satisfied.

F(x + y) = f(x) + f(y)

F(αx) = αf(x)

This is equivalent to requiring the same for any linear combination of vectors.

Occasionaly, V and W can be considered to be vector spaces over different fields. It is necessary in these situations to specify which of these ground fields are being used in the definition of linear. 

Categories within Linear Transformation

Linear Programming

Postings: 1,395

Linear programming is a method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships.


Postings: 691

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Space Constrained Inventories

A grocer has exactly 1,000 square feet available to display and sells 3 kinds of vegetables. The space consumed by each kind of vegetable is proportional to its cost, and tomatoes consume 0.5 square feet per pound. There is a $100 setup cost for replenishing any of the vegetables, and the interest rate is 25% per annum. The 3

Operations Research for Arizona Plumbing and Widgetco

See the attached file. Problem 1 - Arizona Plumbing Arizona Plumbing, which makes, among other products, a full line of bathtubs must decide which of its factories should supply which of its warehouses. Relevant data for Arizona Plumbing are presented in Table 1 and Table 2. Table 1 show, for example, that it costs Arizona

Linear programming for Burger Doodle

This must all be done in Excel. (Very Important) I have also put a link of an example problem for the first part. This how I would like it formatted. Please take a look at it for reference. http://documents.saintleo.edu/docs/AVP/MBA550/MBA550_ch9.mp4 The manager of a Burger Doodle franchise wants to determine how many sa

Linear Programming Using a Demand Function

I am trying to use Linear Programming to solve the attached equation. Accordingly, I used a Demand Function and determined the variable coefficients. I have been receiving comments stating that the problem is unclear and ambiguous. Thus, I tried cleaning it up a bit. The idea is to show the method not a precise answer. So,

Real Life Scenarios: Systems of Linear Equations

Determine a simple real-life example scenario to solve systems of linear equations - Discuss how the math concept applies to your scenario. Scenarios or examples should clearly demonstrate how the math principle can be used to solve a real-life problem. - Research and include references formatted consistent with APA guidel

Formulate the following network problem

(See the attached file for the diagram) - In the above network, a given flow Q is transferred through the network to a demand node N. The goal is to route the flows in such a way that total channel loss is minimized, where channel loss coefficients clij per unit length are given for each link or arc of length Lij. Ignore the

Analytical Models in DSS

A company that assembles electronic alarm systems requires three component parts: C1, C2, and C3. In-house production costs are estimated to be $15 per unit for part C1, $18 per unit for part C2, and $ 20 per unit for part C3. It requires 0.16 hours of machining time and 0.1 hours of finishing time to produce to each unit of pa

Sensitivity Analysis and LP Solve

Transportaton Problem (Minimal Cost) There are three warehouses at different cities: Tauranga, Wanganui and Wellington. They have 180, 100 and 150 tons of paper available over the next week respectively. There are four publishers in Auckland, Palmerston North, Hamilton and Wellington. They have ordered 190, 70, 120 and 50 ton

Linear Mapping in Subsets

Question 1. 1) Suppose (V, | * |) is a normed space. If x, y E V and r is a positive real number, show that the open r-balls Br(x) and Br(x + y) in V are homeomorphic. 2) Suppose V and W are two normed spaces. If A : V ---> W is a linear map, then show that it is continuous at every point v E V if and only if it is continuou

Network Flow using Nodes and Links

A manufacturer must produce a certain product in sufficient quantity to meet contracted sales in the next four months. The production facilities available for this product are limited, but by different amounts in the respective months. The unit cost of production also varies according to the facilities and personnel available. T

LU factorization

Form an LU factorization of the following symmetric matrix to show that it is not positive definite. 4 1 -1 2 1 3 -2 -1 -1 -2 1 6 2 -1 6 1 Using a little ingenuity we can find a non-zero vector such as x^T = ( 0 1 1 0) that does not satisfy the requirement x^T Ax > 0.

Non-Linear Scatterplot

See the attached file. An experiment is conducted to determine the relationship between initial speed and stopping distance of automobiles. A sample of twelve cars is tested and the following data are recorded: Initial speed in mph (x) 20 20 30 30 40 40 50 50 60 60 70 70 Stopping distance in ft (y) 15.9 24 41.2

Schedule Model

Suppose you are waiting in line to check out at a grocery store and there are 7 other customers in front of you (so you are customer 8). By inspecting the amount of items in their baskets, you estimate the following check-out time in minutes: Customer 1 2 3 4 5 6 7 8 Checkout time 10 5 3 7 5 10 2 5 a) What would

Calculating Production and Material Consumption Rates

The Ottawa Coat company manufactures winter coats and spring jackets. The winter coat requires 4m of material while the spring jacket requires just 1m. The company wants to minimize the consumption of material. Each winter coat has one zipper while each spring jacket has two. The seamstresses can sew in zippers at a top rate of

The composition of linear maps and their underlying spaces.

Let U, V, and W be vector spaces over a field F. Suppose that T : U --> V and S : V --> W are linear transformations and that Im(T) = Ker(S). If T is injective and S is surjective, prove that dim(V) = dim(U) + dim(W). Here, dim denotes the dimension of a vector space over the field F.

Isomorphisms Described

f(x)=x^3+x+1 and g(x)=x^3+x^2+1 are irreducible over F_2. K is the field extension obtained by adjoining a root of f and L is the extension obtained by adjoining a root of g. Determine the number of isomorphisms from K to L. (It is not necessary to explicitly describe such an isomorphism)

The Most General Form of a Particular Mobius Transformation

I need a solution for the attached Mobius problem. (a) Find the most general Mobius transformation that maps the right half-plane to the unit disc carrying the point 17 to the origin. (b) Find a Mobius transformation that maps the right half-plane to the upper half-plane carrying the point 7 + 5i to 3i.

Properties of the Dihedral Group D8

Let D_8 denote the group of symmetries of the square. Denote by a a rotation anticlockwise by ?/2 about the centre of the square, and by b a reflection through the midpoints of an opposite pair of edges. (i) Verify that each rotation in D_8 can be expressed as a^i and each reflection can be expressed as a^(i)b, for i?{0,1,2,

ring and kernel

i. Let R be a set with 2 laws of composition satisfying all ring axioms except the comutative law for addition. Use the distributive law to prove that the commutative law for addition holds, such that R is a ring. ii. Find generator for the kernel of the map Z[x]->C defined by x->sqrt(2)+sqrt(3).

Structure of Rings

a) Determine the structure of the ring R obtained from Z by adjoining an element w satisfying each set of relations: (i) 2w-6=0, w-10=0 and (ii) w3+w2+1=0, w2+w=0. b) Let f=x4+x3+x2+x+1 and let y denote the residue of x in the ring R=Z[x]/(f). Express (y3+y2+y)(y5+1) in terms of the basis (1,y,y2,y3) of R.

Correspondence theorem

(a) The kernel of this homomorphism is the principal ideal (x-1). Therefore, Z[x]/(x-1) is isomorphic to Z. According to the correspondence theorem, ideals of Z[x]/(x-1) are in one-to-one correspondence with ideals of Z[x] containing (x-1). Taking into account the above-mentioned isomorphism, we obtain that ideals of Z are in

Problems in Galois Theory

a. Let K be a field of characteristic p > 0, and let c in K. Show that if alpha is a root of f (x) = x^p - x - c, so is alpha + 1. Prove that K(alpha) is Galois over K with group either trivial or cyclic of order p. b. Find all subfields of Q ( sqrt2, sqrt 3) with proof that you have them all. What is the minimal polynomial

Optimize the assignment of customer zones to distribution centers.

There are 5 distribution centers and 8 customer zones with different cost of satisfying demand to different customers from different distribution centers. Minimize the cost of salisfying the demand of all customers zones by assigning them to the distribution centers with some constraines. For detail, kindly see the attached f

Solvable Groups

If G is a finite group, define R = R(G) = INTERSECTION {K < G | G/K is solvable}. a. Show that R is the smallest normal subgroup of G, such that G/R is solvable. b. Show that G is solvable iff R = {1}. c. If H <= G is a subgroup, show that R(H) <= H INTERSECTION R(G). Please see the attachment for question with clear not