Math Help Forum - Linear and Abstract Algebra

Linear Programming - Investment Planning Problem

captjoe — Fri, 20 Nov 2009 21:37:11 GMT

Hello all,

I would appreciate it, if anyone could/would help me to solve a linear programming problem that belongs on the sub-category of investment planning.

The problem:

Airline examines the problem of buying new passenger jet
aircraft to enhance the aviation fleet. Past study
resulted in the selection of three types of aircraft:
Type A: Long-range, with purchase price of EUR 7 million
Type B: mid-range in price 5 million.
Type C: short-range in price 4 million.
It is envisaged that air transport will be large enough for all
distances to the planes of the company, regardless of type, used
essentially the maximum of their capacity. Under these conditions
estimated that the annual net profit, after the service of capital, the
depreciation and all other costs, will be:
0.5 million airplane type A
0.4 million airplane type B
0.3 million for aircraft type C
Also that the company will have enough pilots and staff to
Manning to 30 new aircraft.On the other hand if bought only short-range
aircraft, maintenance crews of the company can serve up to 50
new aircraft. Each aircraft type B, however, employs workshops twice as long
from one aircraft type C, and each plane type A twice as long from one aircraft
type B.
These calculations have resulted from preliminary analysis of
problem. While designed to be more detailed analysis, the administration of
company is now forced to choose between two potential funding.
The company can borrow freely to 100 million or
borrow with government guarantees up to 150 million euros.But In the second case,
State requires for national defense (use the aircraft for
transport in case of war),to purchase at least 5 aircraft type A.
It is assumed that the net profit for the company from each plane is the same
regardless of the mode of financing.
Sought to determine the optimal number of each type of aircraft to be
purchased both in the case of free financing and in
case of financing through government guarantee. It also sought to identify
maximum profits in each case,and choose the most beneficial backing.

Some thoughts, but not sure.

Set, number of type A Aircraft = x1 , number of type B = x2 , number of type C = x3

After that the objective function would be z= x1*0.5 + x2*0.4 + x3*0.3

1st restriction would be x1 + x2 + x3 <= 30
2nd x3<= 50

I can't ''decode'' the ''Each aircraft type B, however, employs workshops twice as long
from one aircraft type C, and each plane type A twice as long from one aircraft
type B.'' part of the problem.(Headbang)

Considering the type of funding I guess it will need to make 2 separate problems for the 100 and 150 million euros types of funding...

Any ideas,anyone?

Dimension of a subspace of R^4

Unenlightened — Fri, 20 Nov 2009 21:25:31 GMT

My question asks me to consider a matrix A

Then find a basis for the nullspace of A, and hence then dimension of the nullspace;
which I find to be .
Since there are two basis vectors the dimension is 2, right?

My problem then is "Using the rank-nullity theorem or otherwise, determine the dimension of the subspace of R^{4} spanned by the four columns of A".

R-N states that dimension null = #cols - rank...
So dimension null =2 , #cols = 4, so I'm guessing rank = 2...

I'm confused by the mention of the "dimension of the subspace of R^{4}" though.
Is the answer they're looking for simply 2? And how ought I to phrase this to explain my answer, rather than merely subtracting one number from another?

Thanks in advance

convergence help

friday616 — Fri, 20 Nov 2009 20:45:13 GMT

Suppose that the summation from k=1 to infinity of a_k converges absolutely. Prove that the series summation from k=1 to infinity of (a_k)^2 converges.

Fields

elninio — Fri, 20 Nov 2009 20:11:18 GMT

Let a,b,c be elements of a field F. Prove that if a=/=0, then the equation ax+b=c has a unique solution.

I just learnt fields and polynomials today. This is the only problem that I cannot solve. (I dont even know where to begin)

Thanks.

S[[x]]' Subring of R[[x]]

lvleph — Fri, 20 Nov 2009 19:47:16 GMT

I am just not confident on a few of the points in the following problems proof and would like some help if it appears I am messed up.

Assume is a ring with subring . Let = \left\{F \in R[[x]]:\; F(n) \in S \; \forall n \in \mathbb{N}\right\}[/math]. Prove that is a subring of .

So i have to prove that
(R1)
(R2)
(R3)
(R4)
(R5) (Added Requirement from Instructor)

(R1) Let then and . However, if and then and , since they are rings. Therefore .

Not confident on the next two.
(R2) Since both are rings we know they have an additive identites, therefore

(R3) Since both are rings we know they have an additive inverse, therefore if and we must have and . So it follows that

(R4) Let then and . Therefore, since both are rings and . Therefore, .

Not very confident on this next one either
(R5) Since both are rings we know they have an multiplicative identites, therefore .

Any comments are welcome. Also, I am told that strictly speaking , but I don't think I completely understand why. It seems to have something to do with the domain and codomain.

minimize operating costs

yomath — Fri, 20 Nov 2009 19:34:02 GMT

I'm a first time user - so I hope I am in the right place. This is a problem that is on a Finite Mathematics Final and no one in the class has been able to solve.

An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class and 1500 econom class passengers. For a certain trip, airplane P1 costs $10,000 to operat and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class and 44 economy passengers. How many of each type of airplane should be used in order to minimize the operating cost.

We initially set up the linear equations and approached it using slack variables.

z = -400x1 - 750x2 - 1500x3
20x1 + 50x2 + 110x3 = 10,000
18x1 + 30x2 + 44x3 = 8,500
x1, x2, x3 > 0

Am I approaching this correctly and can someone help me solve this?

please help!

mhitch03 — Fri, 20 Nov 2009 18:15:48 GMT

I need help with this problem...

Solve the system for x,z in terms of y. Show all row operations.
-x+1/4z=0
x-y+1/4z=0

Thanks!

Domains

johnt4335 — Fri, 20 Nov 2009 17:42:16 GMT

Why is it that R = {a + bsqrt(2): a, b belong to Z} is a domain, but R = { (1/2)(a + bsqrt(2)): a, b belong to Z} isn't?

System to Equations

mhitch03 — Fri, 20 Nov 2009 17:41:52 GMT

Write the system of equations

x+y+z=51
x+2y+3z=88
3x+2y+z=40
x-y+z=18

as a single matrix operation.

Matrix Row Operations

mhitch03 — Fri, 20 Nov 2009 17:40:33 GMT

Could someone help me with this problem? Thanks!

6x+7y=13
x-y=0
x+y=12
(Answer: x=1, y=1)

Solve using matrix operations to get the coefficient matrix into reduced form. Write down the answer.

equation with trigonometric and linear variable

dinNA89 — Fri, 20 Nov 2009 14:52:20 GMT

I wasn't sure where to post this as i'm not sure if it's blindingly simple or more difficult, I'm an out of practice engineering student so either is just as likely
anyway, just trying to solve for theta

I could easily get an answer using a calculator, but i want to do it analytically.

Question about separability

KSM08 — Fri, 20 Nov 2009 11:05:26 GMT

Let K be a field and f a member of K[x] a separable polynomial. Prove that the simple extension M= K[x]/(f) (where (f) is the ideal generated by f) is separable over K. Deduce that if a1,...an are separable elements over K, then the extension K(a1,...,an) is separable over K. Conclude that if f is any separable polynomial, then the splitting field of f over K is separable over K.

Any ideas? I think for the first part I should be using a theorem about the number of homomorphisms from M to a splitting field, but I don't quite see how it fits in...

Many thanks.

Nullspace Q

Unenlightened — Fri, 20 Nov 2009 03:15:49 GMT

The question reads: "For the matrix A below, find a spanning set for its Nullspace.
"

So... I want to solve for such that Ax = 0.

I row reduce; r2 -> r2-2r1, r3->r3-3r1

then r3->r3-r2

Giving
.

So we can say , , , .

So the Nullspace is spanned by

Should Ax not then be zero?

I'm obviously wrong somewhere... but I can't spot where...

symmetric transpose matrix

ecc5 — Fri, 20 Nov 2009 03:07:44 GMT

I am stuck on this problem. Let A be an arbitrary mxn matrix. Show that A transpose times A is symmetric. My teacher has been doing proofs using i and j to denote entries of the matrix, so if you could use that notation when explaining it that would be great. Thanks!

atomic factorization

knguyen2005 — Thu, 19 Nov 2009 22:44:34 GMT

Let m in Z, where m is not a square in Z
Define N: Z[sqrt(m)]---> N
Suppose that : a belongs to Z [sqrt(m)]. If N(a) is a prime in N . Show that a is an atom in Z [sqrt(m)]. Give an example of an m and a in Z [sqrt(m)] such that a is an atom but N(a) is not a prime

Can you give me some hints please?

Thank you