Finding the area of a triangle

We begin by asking you to recall that moment not so long ago when you were in an algebra or geometry class, and a problem might begin when we gave you the coordinates of three points in the coordinate plane, A=(2,1); B=(5,6) and C= (9,-1). Among the myriad things we might have asked you to do with this given information would be two that are easily answered by the methods of determinants; “Find the area of the triangle formed?” and “Prove that the three points are, or are not, collinear (lie on the same line).”

Finding the area of a triangle

Taking the points in pairs we could find the length of each segment, and then employing the well known formula of Heron (you do remember Heron’s Formula, don’t you?) to find the area. We would arrive at an area of 20.5 square units. But now that you know how to evaluate the determinant of a matrix, we can instead just enter the x values of each point as one column of the matrix, enter the y values as a second column, and add a third column of all ones (someday I will explain about cross products of vectors and this third column will seem more clear). We get the matrix shown at right. And if we evaluate the determinant of the matrix we get –41, as the screen capture shows. Two things may concern you; but if you are at all clever, you realize that the area 20.5 is ½ the determinant of –41 except for the sign.

At this point I remind you that the sign of the determinant depends on the order we enter the values, and if we switch the 1^st and 2^nd rows, we get a positive determinant. So if we can suspend the concerns and accept that the area of the triangle formed is ½ the absolute value of the determinant, we have hit upon a easy, efficient, way to find the area of a triangle using technology and our new friend the determinant. (Now is when you say “Wow, way cool!”)

But what about the other problem of proving that the three points lie in a straight line. Well, if they were in a straight line, the area of the triangle would be zero, and the determinant would be zero, and that would tell us. If the points are NOT in a straight line, the determinant will NOT be zero because any three points in a plane which are NOT collinear, form a triangle with a non-zero area. So the same determinant answers both questions.

Business Math

I got this example from this link http://www.mff.cuni.cz/veda/konference/wds/contents/pdf06/WDS06_106_m8_Ulrychova.pdf

in this example there are three people (P1, P2, P3) who produce three commodities (z1, z2, z3)

Using Matrix Functions For Finance

Source:http://www.mathworks.com/access/helpdesk/help/toolbox/finance/f2-1001993.html

Many financial analysis operations involve sets of numbers; Matrices, matrix functions, and matrix algebra are the most efficient ways to analyze sets of numbers and their relationships. Spreadsheets focus on individual cells and the relationships between cells. A matrix-oriented tool like MATLAB software manipulates sets of numbers more quickly, easily, and naturally.

For example, here is a 2-by-3 matrix of two bonds (the rows) with different par values, coupon rates, and coupon payment frequencies per year (the columns) entered using MATLAB notation:

Bonds =

[1000 0.06 2
500 0.055 4]

Vector. A matrix with only one row or column. The description is always "row-by-column." For example, here is a 1-by-4 vector of cash flows in MATLAB notation:
Cash = [1500 4470 5280 -1299]

Referencing Matrix Elements
To reference specific matrix elements, use (row, column) notation. For example:

Bonds(1,2)
ans =
0.06

Cash(3)
ans =
5280.00

We can also enlarge matrices using small matrices or vectors . For example,

AddBond = [1000 0.065 2];
Bonds = [Bonds; AddBond]

addsanother row to the matrix and creates

Bonds =

1000 0.06 2
500 0.055 4
1000 0.065 2

Transposing Matrices
In MATLAB, the apostrophe or prime character (') transposes a matrix: columns become rows, rows become columns. For example,

Cash = [1500 4470 5280 -1299]'
produces
Cash =

1500
4470
5280
-1299

Business Math

Three people denoted by P1, P2, P3 intend to buy some rolls, buns, cakes and bread. Each of them needs these commodities in differing amounts and can buy them in two shops S1, S2. Which shop is the best for every person P1, P2, P3 to pay as little as possible? The individual prices and desired quantities of the commodities are given in the following tables:

Demanded quantity of foodstuff: Prices in shops S1 and S2:
Roll bun cake bread

P1 6 5 3 1 S1 S2
P2 3 6 2 2 Roll 1.50 1.00
P3 3 4 3 1 Bun 2.00 2.50
Cake 5.00 4.50
Bread 16.00 17.00
For example, the amount spent by the person P1 in the shop S1 is:
6 · 1.50 + 5 · 2 + 3 · 5 + 1 · 16 = 50
and in the shop S2 :
6 · 1 + 5 · 2.50 + 3 · 4.50 + 1 · 17 = 49,
for the other people similarly. These calculations can be written using a product of two matrices


6 5 3 1
P= 3 6 2 2
3 4 3 1
(the demand matrix) and
1.50 1
Q= 2 2.50
5 4.50
16 17
ULRYCHOV´ A: SEVERAL SIMPLE REAL-WORLD APPLICATIONS OF LIN. ALG. TOOLS
(the price matrix). For example, the first row of the matrix


50 49
R = PQ = 58.50 61
43.50 43.50

expresses the amount spent by the person P1 in the shop S1 (the element r11) and in the shop
S2 (the element r12). Hence, it is optimal for the person P1 to buy in the shop S2, for the
person P2 in S1 and the person P3 will pay the same price in S1 as in S2.

http://www.mff.cuni.cz/veda/konference/wds/contents/pdf06/WDS06_106_m8_Ulrychova.pdf

Tarek Geagea
NDU

Cryptology

To encode a short message a number can be assigned to each letter of the alphabet according
to a given table. The text as a sequence of numbers will be organized into a square matrix A;
in the case that the number of letters is lower than the number of elements of the matrix A,
the rest of the matrix can be filled with zero elements. Let a nonsingular square matrix C be
given. To encode the text the matrix A can be multiplied by the matrix C for example on the
left. Let the following table and the matrix C be given:

A B C D E F G H I J K L M N O P Q R S T U V W X y Z
8 7 5 13 9 16 18 22 4 23 11 3 21 1 6 15 12 19 2 14 17 20 25 24 10 26

C =

2 0 1
1 0 1
0 1 0

We put the text ”BILA KOCKA” (a white cat) into the matrix A:
A =

7 4 3
8 11 6
5 11 8

and encode the text:
Z = CA =

19 19 14
12 15 11
8 11 6

.
To decode the message we have to multiply the matrix Z by the matrix C−1 on the left:
C−1Z =

1 −1 0
0 0 1
−1 2 0


19 19 14
12 15 11
8 11 6

= A.
Since the matrix multiplication is not commutative, it is necessary to keep the order of
the matrices in the product. If we multiply the matrices C−1 and Z in the opposite order, we
obtain

ZC−1=

19 19 14
12 15 11
8 11 6



1 −1 0
0 0 1
−1 2 0

=

5 9 19
1 10 15
2 4 11

and it means ”CERNY PSIK”(a black dog).

FIRAS ABDEL DAYEM
http://www.mff.cuni.cz/veda/konference/wds/contents/pdf06/WDS06_106_m8_Ulrychova.pdf

Traffic Networks

Many network problems can be modeled by systems of linear algebra, for example traffic networks.Nowadays, people face traffic problems everyday for different reasons like accidents, construction,... etc. Making a system that correspondes to a certain block of streets, and solving it, will help us by showing us the changes that should be done in order to avoid negative flow.This is an example of how linear algebra can be useful in real life. http://aix1.uottawa.ca/~jkhoury/networks.htm
Tarek geagea
ID# 20081699

RSS FEED

Hello guys, well, on the right hand side of the newest posts, you can see "Subscribe via Email", delivered by FeedBurner. This means, that if you enter your email in that box, then ANY new post or any new event happening in the Blog, you will get an email that keeps you up-to-date with the new activity.


Hence, if you do not want to keep checking the Blog to look for new activities, you can subscribe to FeedBurner and you will get these changes via email! :)


Nice, eh? Why don't you try it out, if you wish.  Zeina

LAU students are checking your Posts and they will COMMENT!!!

Dear NDU students,

I invited LAU (Byblos) students (mainly Biology and Pre Med Students) to your BLOG! They will comment on your wonderful posts by Thursday, January 21, 2010!
Check it out and respond if you would like :)

Wait until Tuesday, Jan. 19 though. ok??

Cheers, Zeina

Interesting Math Website

Hello!
I just found an interesting Math Site that can help you while studying and can give you lots of insights and new info about Linear Algebra!
Check it out!
This is it: http://www.intmath.com/Matrices-determinants/Matrix-determinant-intro.php
Zeina

The stress matrix

Surface tractions, or stresses acting on an internal datum plane, are typically decomposed into three mutually orthogonal components. One component is normal to the surface and represents direct stress. The other two components are tangential to the surface and represent shear stresses.

Direct stresses tend to change the volume of the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus (which depends on the Young's modulus and Poisson ratio). Shear stresses tend to deform the material without changing its volume, and are resisted by the body's shear modulus

the stress state at point P can be represented by an infinitesimal cube with three stress components on each of its six sides

nine stress components from three planes are needed to describe the stress state at a point P.

These nine components can be organized into a matrix as seen in the picture.

As a mechanical eng. student at NDU i wanted to found something that relates matrices with eng. application ,that is why i did my post on stress matrices.

site used: http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/stress.cfm

Elie Kassab

Interested In Music? Have fun with Matrices

Markov chains (a discrete random process with the Markov property) are employed in algorithmic music composition, particularly in software programs such as CSound or Max. In a first-order chain, the states of the system become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce and output note values based on the transition matrix weightings, which could be MIDI note values, frequency (Hz), or any other desirable metric.


A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of phrasal structure, rather than the 'aimless wandering' produced by a first-order system.
URL: http://en.wikipedia.org/wiki/Markov_chain#Music
Who don't like music? No matter which genre? We all like it!!!
Patrick Daou.

When done with blogging

Hello again,


Next week, when done with blogging, and after you finish your exam 2, we should keep what we started with this Blog!


Give me ideas......


One idea I have to OPEN the BLOG's COMMENTS to the public and see what people around, students in Lebanon and abroad will comment on your posts! BUT you will have to keep checking your Blog OR USE SOMETHING CALLED RSS FEED! do you know what it means??? 


If yes, explain in a comment :)


The BLOG also will be shared, starting next week with the other sections of Linear Algebra at NDU and in a different American University as well!!!!!!!!!!!!


Thanks to you who worked hard!


So, WHAT IS THE RSS FEED?????????? 


Zeina

matrices in real world

we can know from this example how marix ca be usefull for some one who has his own store and wants to do his own sales.
Matrix operations are frequently used in real world computations.
Suppose a matrix gives the number of different computers that Michael sells in the first two months at his new job. He sells 30 units of computer Brand X, 21 units of computer Brand Y,
and 11 units of computer Brand Z. These sales numbers have been entered in matrix [A].
The row gives the number of sales for each brand and the columns represent the
brand of computer. The name of this matrix is matrix [A] and its dimension is 1 by 3 (1 x
3). This means that the matrix has one row and three columns.
Michael sells a Brand X computer for $999.95, a Brand Y computer for $875.50,
and a Brand Z computer for $1059. Matrix [B] gives the prices of the three brands of
computers. This time, the rows give the brands and the column gives the price of each
type of computer. The dimension of matrix [B] is 3 x 1.
Michael gets a commission of 12% on each computer he sells. Multiply matrix
[B] by 0.12 to compute the commission on each brand of computer.
How much is his total commission? Store (STO key) the results of the
multiplication to a new matrix, [C]. Multiply matrix [A] (the row gives the number of
computers and the columns give the type of computer) by matrix [C] (the rows give the
type of computer and the column gives the commission on each type) to obtain the total
commission for this number of computer sales. The product of a 1 x 3 matrix and a 3 x 1
matrix is a 1 x 1 matrix (one row and one column). His total commission is $7,203.96.

Naji Achkar mat 215 course linear algebra
application:wiki.answers.com/.../Applications_of_matrix_in_real_life

Encrypting Text Using Linear Algebra

Each letter is first encoded as a number. Often the simplest scheme is used: A = 0, B =1, ..., Z=25, but this is not an essential feature of the cipher. A block of n letters is then considered as a vector of n dimensions, and multiplied by a n × n matrix, modulo 26. (If one uses a larger number than 26 for the modular base, then a different number scheme can be used to encode the letters, and spaces or punctuation can also be used.) The whole matrix is considered the cipher key, and should be random provided that the matrix is invertible in (to ensure decryption is possible).
Consider the message 'ACT', and the key below (or GYBNQKURP in letters):

Since 'A' is 0, 'C' is 2 and 'T' is 19, the message is the vector:

Thus the enciphered vector is given by:

which corresponds to a ciphertext of 'POH'. Now, suppose that our message is instead 'CAT', or:

This time, the enciphered vector is given by:

which corresponds to a ciphertext of 'FIN'. Every letter has changed. The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at once.
Article found at http://en.wikipedia.org/wiki/Hill_cipher
Alexander Abi Chaker 20091978

Real-world Application of Linear Algebra

The Earth travels around the Sun in an ellipse, and the Sun is at a focal point of that ellipse. Likewise for a satellite travelling around the Earth (Figrue 1). It is possible for a satellite to travel in a circular orbit, but that is a special case. From such information we are able to derive the orbit of an object using linear algebra. The satellite-based Global Positioning System
(GPS) works by locking on to the signals of at least three satellites to calculate a 2 dimensional
position (latitude and longitude) and track movement. With four or more satellites in view, the receiver can determine the userʼs 3D position (latitude, longitude and altitude). The location
of an object could be identifi ed using 3 satellites or more. This can be done by solving linear
equations.
Mixing chemicals is a method used by a chemist to provide a useful outcome (e.g. solid, gas, liquid, spark, perfume... etc.) or not useful outcome (e.g. fire, bad smell, explosion... etc.). Dealing with chemicals is dangerous, but mixing chemicals is even more dangerous. The chemist has to
ensure that s/he is using the right amount of each chemical or there may be a disaster (or by luck a new invention). For example, when developing photographic negatives, s/he is dealing with three chemicals (developer, fixer and stop path), each one to be mixed with water in amounts specified by the manufacturer. Incorrect mixing will spoil your nice negatives. Where there is a large number of chemicals involved, computer packages can help to determine the right amounts to be included in the mix. Those computer packages use a technique that involves linear algebra.

Figure 1. Satellite communications and the satellite-
based Global Positioning System (GPS) have
benefi tted from mathematical models using linear
algebra.


Figure 2. Chemical mixing can be modelled using
linear algebra.
http://physics.dit.ie/resources/yearbook2005/alkalbani.pdf

Working with matrices in MATLAB

MATLAB has many useful commands for creating,manipulating or operating on matrices.I hope to show you just a small sample here just to show how can matlab create any matrices.
You can create a 3 .4 matrix full of zeroes:
» A = zeros (3,4)
A=
0 0 0 0
0 0 0 0
0 0 0 0
or u can create a martrices full of ones:
» B = ones (3,4)
B =
1 1 1 1
1 1 1 1
1 1 1 1
Note that you can make a row vector full of zeros by typing zeros (1,5) or a column vector of zeros using zeros (5,1).Typing zeros (5) will give a 5.5 matris full of zeros.Perhaps more or a useful than a matrix full of ones is the identity matrix which have ones on the main diagonal and zeros elsewhere.Remember that identity matrices are always square,ao that you only need to specify the number of rows in the matrix:
» I = eye (4)
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Identity rows and columns of matrices
The 3rd row of a matrix can be seen in Matlab by typing:
» I(3,:)
ans =
0 0 1 0
The 3rd column can be seen i a similar fashion:
» I(:,3)
ans =
0
0
1
0
You can even select submatrices in this way.Here we pick off a 3x3 matix.
» I (1:3,2:4)
ans =
0 0 0
1 0 0
0 1 0
DIAGONALS AND TRIANGLES
i created a 4x4 random matrix:
» A = rand (4)
A =
0.9501 0.8913 0.8214 0.9218
0.2311 0.7621 0.4447 0.7382
0.6068 0.4565 0.6154 0.1763
0.4860 0.0185 0.7919 0.4057
you can select the upper part of A using
triu:
» upper = triu(A)
upper =
0.9501 0.8913 0.8214 0.9218
0 0.7621 0.4447 0.7382
0 0 0.6154 0.1763
0 0 0 0.4057
The lower triangular part may be picked off using
tril:
» lower = tril (A)
lower =
0.9501 0 0 0
0.2311 0.7621 0 0
0.6068 0.4565 0.6154 0
0.4860 0.0185 0.7919 0.4057
The main diagonal may be extracted (as a vector) using
diag:
1
» d = diag(A)
d =
0.9501
0.7621
0.6154
0.4057

using matrices to solve optimization problems

i found this example on this link:
http://en.wikipedia.org/wiki/Linear_programming#History_of_linear_programming

suppose that a farmer wants to plant his land (of area A) with either wheat or barley or a combination of both. Moreover he has a limited allowed quantity of fertilizer (F) and insecticide (P) which is used in different amounts per unit area for wheat (F1, P1) and barley (F2, P2). the farmer wants to know what amounts should he plant of wheat and barley in order to maximize his profit.
let S1 be the selling price of wheat and S2 the selling price of barley.
this maximization problem can be solved by matrices


matrices are a easy way to solve optimization problems especially when there is a big number of variables.

EIGENFACES AND DIGITAL IMAGE RECOGNITION

David Mumford, Alan Yuille, and Peter Hallinan at Harvard have taken a different tack in the computer matching of faces. One problem in analyzing a picture of a person's face is that the image depends largely on the presence or absence of shadows, which depend, in turn, on lighting conditions. Mumford and Hallinan address this by computing what they call "eigenfaces".
To obtain these, they photograph a subject under as many as a hundred different lighting conditions. Once digitized, these images can be interpreted as points in a very high-dimensional space (the dimension is equal to the number of pixels in the image). The resulting cloud of points is shaped something like an ellipse. The axes of the ellipse and their lengths can be determined by a computation in linear algebra (to be precise, by computing the eigenvectors and eigenvalues of the Matrix IIt where I is a column-by-column listing of the images and It is the transpose of I). The eigenfaces lie along these axes. The idea is to reconstruct a good approximation to a face under general lighting conditions by combining just a few eigenfaces and then "warping" the result.
The theory of linear agebra carries over into the high-dimensional world of digital images with the computation of "eigenfaces" in an application of control theory to an important problem in pattern recognition. (Figure courtesy of Harvard Robotics lab.)
(From: The Gentle Art of Control , in "What's Happening in the Mathematical Sciences", Vol. 3, by Barry Cipra, and published and available from the American Mathematical Society .)

Using matrices to Enhance Marketing

Much of marketing involves connections. Connecting with clients and prospects. Making connections between a client’s needs for legal services and the services your firm can provide. Connecting a legal service with industry knowledge and professional contacts to enhance the breadth and depth of your practice. Building trust and strengthening relationships with clients.

Much of marketing strategy involves connectivity. Seeing and understanding the complex relationships between lawyers and clients, between value and client loyalty, between supply and demand, between recurrent and non-recurrent client needs for services, between referral sources and end-consumers, and the list goes on.

Connectivity can certainly be identified on a case-by-case basis. Many successful lawyers have good understanding of connectivity, know how to get from A to B, and succeed at developing long term business by making it up as they go along.

However, to develop marketing organizations that are effective and successful—whether at the law firm, practice group or marketing team level—common goals, strategies, targets and vehicles must be formulated and implemented. And, in order to develop organizational marketing programs and strategies, case-by-case connectivity must be converted into patterns that can be systematically identified, evaluated, prioritized and acted upon.

One simple-yet-creative tool, and one that is generally underutilized in marketing legal services, is the matrix. Properly constructed and employed, matrices can be of great assistance in converting random information into meaningful, useful patterns. These patterns can greatly assist in developing more targeted, focused marketing strategies. Patterns can also be derived to assist organized follow through and management of strategies. In addition, while somewhat non-traditional compared with the goals-first-and-then-strategy planning continuum, conclusions drawn from matrices can help test, shape, or re-shape a firm or practice group’s marketing goals and objectives.

The following examples of marketing matrices illustrate their diverse, pragmatic application to marketing, including market research, targeting, planning, strategizing, and implementation.
Client Service Grid

This matrix can either be used with existing clients or prospects. It is a method of channeling preliminary knowledge and research into the formation of cross-selling or entry strategies for individual clients.
Client/Prospect

Real Estate

Environmental

Construction

Other Practice Areas
Client A

Uses our firm

Can use our firm

X

Client B

Uses our firm

X

Uses our firm

Prospect A

Uses our firm

Can use our firm

Not sure


Based on assumed information, a cross-selling opportunity appears with Client A in the environmental area. Client B warrants a retention strategy, assuming the firm handles all of the client’s work in these areas. Prospect A appears to have potential in real estate and environmental, and may have potential in construction law.
Key Client Matrix

This matrix is a vehicle to objectively identify a law firm’s key clients. The assumption underlying this matrix is perhaps most important: a key client has at least one of the following characteristics: current business volume; potential business volume; or value as a referral source. In other words, a client with great potential for legal work, or proven value as a referral source, may be regarded as a key client even if the current revenues from that client are modest.
Client

Current Fee Revenue

Potential Revenue

Value as Referral Source
Client A

Major

X

Maybe – not yet
Client B

Small

Major

Maybe
Client C

Small

Small

Limited

Even this simple example suggests more than a process for identifying key clients. Any key client may be targeted for special client service initiatives, marketing planning, or other attention. Client A may be an untapped source of referrals, depending on the industry, client decision-makers and other factors. Client B suggests expansion-oriented objectives and strategies, through more of the same type(s) of legal services, or through diversified services via strategies for cross-selling or team-selling.
Client/Office Matrix

This matrix was developed to help multi-office law firms systematically identify clients with business operations in more than one geographic area. This matrix illustrates the beginning, culminating with potential targets and synergies. Once identified, individual client’s legal needs, the corresponding capabilities of the firm/offices to deliver services, client decision-making structures for retention of counsel, competition, and other factors would need to be assessed prior to and during strategy development. The targeting information can be used:

* To introduce the lawyers and services of one or more additional offices directly to the client for local services;
* To expand overall representation by team-selling two or more offices;
* To cross-sell specialized lawyers from another office into a branch office locale.

Client

Smithville Office

Jonesdale Office

Bigcity Office
Client A Operations in Smithville; we provide services Operations in Jonesdale; we don’t provide services Operations in Bigcity; we don’t provide services
Client B Operations in Smithville; we don’t provide services No operations Operations in Bigcity; we provide services

Regarding Client A, the firm can explore direct introductions or cross-selling the lawyers and services by offices in both Jonesdale and Bigcity areas. In addition, if we assume for this illustration that the law firm’s largest office is in Bigcity, there may be specialty cross-selling of Bigcity Office lawyers and services for Smithville-based client legal service needs. To the extent other Smithville and Jonesdale offices have specialists, the latter strategy would be applicable for them as well.

Client B may have locally based needs in Smithville, warranting introductions of Smithville lawyers and service capabilities. Smithville Office specialists can be cross-sold. In general, the firm’s local presence can be touted as a commitment to the community, local offices can be used for Client B meetings and even offered to Client B for off-site meetings.
Contact Priority Matrix

Referral sources for legal work can be affirmatively targeted for development. First, profile an ideal client for a practice group or specialty. Second, think of groups, professions, or individuals who are routinely “one step away” from the ideal client—individuals who are likely to come into contact with the ideal clients. For example, in the tax controversy area, accountants and accounting firms would be “one step away”—connected routinely to individuals and businesses, some of which, at any given time, may be about to experience a tax controversy requiring a lawyer’s help. Accounting firms, or more likely accountants, can be identified and listed—step three.

At this point, the list lacks prioritization and may be long and unwieldy. In general, establishing strong, repetitive referral sources involves establishing relationships—usually more is not merrier. Besides, substantial time can be involved in cultivating referral sources. If the time is devoted to superficially following up on a long list of only somewhat interesting or somewhat interested referral sources, it is likely to be time squandered.

The Contact Priority Matrix was developed to help easily prioritize a list of potential referral sources, for expansionistic pursuit. When potential referral sources have been listed, the Matrix below provides two rating scales to be applied to each potential referral source. Each scale decreases from 3 to 1. The left-hand scale rates each potential referral source based on the quality of their potential client contacts. Assign a 3 if the referral source definitely has the contacts; a 2 if it is a maybe, for example, a referral source on the way up; and a 1 if it is doubtful, for example the source has turned to technology consulting rather than tax work. The right-hand scale describes the strength of the current tie with the prospective referral source—the propensity to help. Again the scale descends from a 3, (“Definitely, since we represented him personally, he thinks the world of our firm”); to a 2 (“I’m not sure, we’re cordial, but haven’t really talked about what we do”); to a 1 (doubtful).
Referral Source

Does the potential referral source have contacts of interest to us?

Is the potential referral source willing to help or work with us?
Potential Source A 3 – Definitely 3 – Definitely
Potential Source B 3 – Definitely 1 – Doubtful
Potential Source C 2 – Maybe 3 – Definitely

Add the two scores of each source, (e.g.: Source A equals 6; Source B equals 4). Contact those with combined scores (i.e., priority) of 6, then 5, then 4, etc. With limited time and a desire to focus on high potential, willing referral sources, those on the list receiving 6’s would be contacted first, followed by those receiving 5’s, possibly followed by those receiving 4’s.
Conclusion

For many, if not most, law firms, the next level of marketing achievement is to become effective, successful marketing organizations—at firm, department, office, and specialty team levels. Firms that become effective marketing organizations augment individual entrepreneurial efforts by creating more opportunities, and by better recognizing and capitalizing on the opportunities created.

The law firm as an institution and individual lawyers can help build an effective marketing organization, by thinking strategically—visualizing the interrelationships and connectivity between factors and forces in the form of patterns, as well as individual fact situations. From this foundation, law firms can begin intuitively perceiving and evaluating the potential of interrelationships, to ultimately, anticipating and creating the connections that yield new prospects, clients, service vehicles, marketing tools—and even new services.

Matrices are simple vehicles that can be used, creatively and with broad application, to assist law firm market research, targeting, planning, strategizing and implementation.

Hello, this problem is an electronics application of 3×3 System of equations. we should solve the matrix equations arising from this circuit.(N.B: use the kirchhoff's Law)
The website is: http://www.intmath.com/Matrices-determinants/6_Matrices-linear-equations.php
example 4 in the list of examples...hope to be a good example.

Using Inverse Matrices in Real Life

i think this is very interesting!!
we can use matrices to create a cryptogram
cryptogram is a message written according to a secret code.
From greek word kryptos meaning hidden and gramma meaning letter.
steps 2 create a cryptogram:
. Assign a number to each letter in the alphabet with out a blank space.
. Convert the message to numbers partitioned into 1*2 uncoded row matrices.
. To encode a message choose a 2*2 matrix A that has an inverse and multiply the uncoded row matrices by A on the right to obtain coded row matrices.
for example:
convert : GET HELP
. 7 5 20 0 8 5 12 16
. [7 5] [20 0] [8 5] [12 16]
. encode use A= 2 3
-1 -2
to see the answer just download the following:
http://www.google.com/url?sa=t&source=web&ct=res&cd=11&ved=0CAcQFjAAOAo&url=http%3A%2F%2Fguilford.rps205.com%2Fdepartments%2Fmath%2FLinks%2FAlgebra%2520II%2520-%2520College%2520Alg%2F4.4%2520Cryptograms%2520Using%2520Inverse%2520Matrices%2520in%2520Real%2520Life.ppt&ei=4bJDS7q6Nc7X4gap5sTGDA&usg=AFQjCNER0oUlU2FsTOVJ0Nuam0xTfjPIeg&sig2=UYWzPn9H0M5WADqSk0NopA

we can encode and decode!!!



Claudia Abdel-Sater

Matrices
In the Real World
Any data that is organized in columns and rows can be represented as a matrix. Matrices are used to represent real-world data such as the habits, or traits of a population of people.

http://www.mathwarehouse.com/algebra/matrix/real-world-applications-matrix.php

raed abdel samad.

Matrices are used in economics

Hello all,
All of us sooner or later will be married. And all of us will have children,which means having a family. However having a family means spending a lot of money on its expenses. Therefore you should know how to find an easy way that'll help you know how much you have spent money on the expenses. The most wonderful way is using the matrices. If you don't believe me you can check this example which you can find by visiting
http://www.sosmath.com/matrix/matrix0/matrix0.html
Aida Lteif

Vectors and matrices

My Research is found at: http://www.devmaster.net/wiki/Vectors_and_matrices

This research explains how to use matrices in 3-dimensional space and that could be explained by explaining vectors that we already started earlier today in class, it shows us how to add or subtract two vectors which is just like ordinary numbers, and how to multiply a vector product by a scalar.

Here are two examples how to add or subtract two vectors.

• You can add two vectors by adding their coordinates. Geometrically, thinking of vectors as arrows, this corresponds to taking the tail of one arrow and putting it at the head of the other arrow. So if the vector a points from P to Q, and b points from Q to R, then the combined vector a + b points from P to R.

• You can also subtract two vectors, by subtracting their coordinates. Geometrically, this corresponds to moving both vectors so that they start at the same point, and drawing the vector that connects their end points. So if a points from P to Q, and b points from P to R, then b - a points from Q to R.

What did i learn:

i learned how to use the matrix library in the bottom of the article, that i could use in my C++ courses.

Jose

The Leontief open Model

Example:
Consider an open economy with three industries: coal-mining operation, electricity-generating plant and an auto-manufacturing plant. To produce $1 of coal, the mining operation must purchase $0.1 of its own production, $0.30 of electricity and $0.1 worth of automobile for its transportation. To produce $1 of electricity, it takes $0.25 of coal, $0.4 of electricity and $0.15 of automobile. Finally, to produce $1 worth of automobile, the auto-manufacturing plant must purchase $0.2 of coal, $0.5 of electricity and consume $0.1 of automobile. Assume also that during a period of one week, the economy has an exterior demand of $50,000 worth of coal, $75,000 worth of electricity, and $125,000 worth of autos. Find the production level of each of the three industries in that period of one week in order to exactly satisfy both the internal and the external demands.

It's interesting to see how matrices are being implemented in industries and economy.
You can see the solution of this example on this link: http://aix1.uottawa.ca/~jkhoury/leonteif.htm


Cynthia Azar 20071375

DEADLINE

Dear all,

Today is Sunday, Jan 3, 2010 and tomorrow, first day of school is the DEADLINE for submitting a Post and 1 comment. 

For the others, from this minute on, the students who missed the deadline and/or the students who did not get a positive comment from me:

 I will give an additional few days more: until Friday, January 8, 2010, 8 am 

BUT you need to POST AND COMMENT ON 3 POSTS INSTEAD OF ONE!!!!!

BLOGGING IS GOOD! :)

See you tomorrow!

Zeina

Using Matrices Functions for Finance

Many financial analysis procedures involve sets of numbers;.Matrices, matrix functions, and matrix algebra are the most efficient ways to analyze sets of numbers and their relationships. A matrix-oriented tool like MATLAB software manipulates sets of numbers more quickly, easily, and naturally.
For example, here is a 2-by-3 matrix of two bonds (the rows) with different par values, coupon rates, and coupon payment frequencies per year (the columns) entered using MATLAB notation:

Bonds = [ 1000    0.06  2

                     500  0.055  4 ]


Cash = [ 1500  4470  5280  -1299 ]


Referencing Matrix Elements

To reference specific matrix elements, use (row, column) notation. For example:

Bonds(1,2)

ans =

0.06

Cash(3)

ans =

5280.00

You can enlarge matrices using small matrices or vectors as elements. For example,

AddBond = [ 1000   0.065   2 ];
Bonds = [ Bonds;  AddBond ]

Adds another row to the matrix and creates

Bonds =


1000     0.06  2
  500   0.055  4
1000   0.065  2



Transposing Matrices

Sometimes matrices are in the wrong configuration for an operation. In MATLAB, the apostrophe or prime character (') transposes a matrix: columns become rows, rows become columns. For example,

Cash = [ 1500 4470 5280 -1299 ]'

produces

Cash =

  1500
 4470
 5280
-1299

It's interesting to see matrices being used in this line of work.
More information can be found at this site

Richard Khoury

BASSAM EL KHOURY

THIS IS A VERY STANDARD EXERCISE IT WAS A MIDTERM EXAM IN AN AMERICAN UNIVERSITY BUT LOOK FOR NUMBER 2 IT IS SOMETHING INTERESTING AND NEW IN MATRICES!
I HOPE IT WILL BE USEFULL FOR YOU!

MY COMMENTS FOR OTHER POSTS:

VERY INTERESTING EXERCISES I SAW NEW STUFFS I DIDN'T KNOW IN MATRICES AND WE CAN SAY THAT LINEAR ALGEBRA SEEMS TO BE A VERY LARGE MATHEMATICAL DOMAIN!

Matrices used by the Input-Output model in Economics

Hello everyone!
My post focuses on how matrices can be used in economics, specifically in the input-output model. It relates the columns of the matrices with the value of the inputs of the industries, and the rows with the value of the outputs of the industries, as shown below.

In economics, an input-output model uses a matrix representation of a nation's (or a region's) economy to predict the effect of changes in one industry on others and by consumers, government, and foreign suppliers on the economy.Input-output depicts inter-industry relations of an economy. It shows how the output of one industry is an input to each other industry. Leontief put forward the display of this information in the form of a matrix. A given input is typically enumerated in the column of an industry and its outputs are enumerated in its corresponding row. This format, therefore, shows how dependent each industry is on all others in the economy both as customer of their outputs and as supplier of their inputs. Each column of the input-output matrix reports the monetary value of an industry's inputs and each row represents the value of an industry's outputs. Suppose there are three industries. Column 1 reports the value of inputs to Industry 1 from Industries 1, 2, and 3. Columns 2 and 3 do the same for those industries. Row 1 reports the value of outputs from Industry 1 to Industries 1, 2, and 3. Rows 2 and 3 do the same for the other industries.
Below is a direct example:
Unfortunately, the blog does not read the format of the matrices, so here is the link:http://en.wikipedia.org/wiki/Input-output_model
Check it out yourselves.

The best thing about my application is that the mathematics of the input-output model is straighforward eventhough its requirements are enormous because all the costs, revenues, and expenditures of each brand of economic activity has to be represented.
As a result, not all countries collect the required data and data quality varies, eventhough a set of standards for the data's collection has been set out by the United Nations through its System of National Accounts(SNA): the replacement for the current 1993 SNA standard is pending. Because the data collection and preparation process for the input-output accounts is necessarily labor and computer intensive, input-output tables are often published long after the year data was collected--typically as much as 5-7 years after. Moreover, the economic "snapshot" the benchmark version of the tables provide of the economy's cross-section are taken only once every few years, at best. Although many developed countries estimate input-output accounts annually and with much greater recency.

At least we spot relations between matrices/mathematics and real life.
HOPE YOU LIKE IT!!!
Rolland Al-Ahmar 20091284

The Uses of Matrices

There are many uses for matrices in everyday life. One example of how matrices can be used is with encryption. When a programmer encrypts or codes a message, he can use matrices and their inverse. The internet function could not function without encryption, and neither could banks since they now use these same means to transmit private and sensitive data. Many geologists make use certain types of matrices for seismic surveys. Matrices are also used in graphs and statistics for doing scientific studies in many other different fields. Matrices are used to calculate gross domestic product in economics, and thereby help in calculation for producing goods more efficiently. Matrices are also sometimes used in computer animation. They can also be used as labels for students to stay organized. They could label things like “School,” “Sports,” “Home work,” and “Recreation.” Along the side, list yourself and 3 friends. Poll your friends and fill out the table, rounding the time to whole numbers. Compare this matrix to the matrices you have been doing. Like, in the example shown previously, matrices are useful for polls. Matrices are very useful for organization, like for scientists who have to record the data from their experiments if it includes numbers. In engineering, math reports are recorded using matrices. And in architecture, matrices are used with computing. If needed, it will be very easy to add the data together, like we do with matrices in mathematics. Like in some problems of our homework, matrices could be useful to figuring out things like price and quantity, like with the foods and prices in our homework. As you can see, there are many and very useful ways matrices could be applied in our everyday lives and even in the future.

thank you.

i was doing my math revision in this holiday and while reading this blog i really got important informations about many things i didnt know especially about determinants. thank you!!

MATRICES IN CODING MESSAGES

http://www.sosmath.com/matrix/coding/coding.html
This link contains the most important application of matrices in the coding pattern!!!!
after viewing this link really we can say that matrices are related in our real life.they are used for sending text messages which is so important.
In this post: to encrypt a message we convert the msg to a matrix BUT to decrypt the msg we need to know the inverse of the matrix !
All what we learned in class including row operations,determinants,upper triangular matrix are applied to be able to sent the message finally!
hope u enjoyed these nice information that teach us how to solve a problem through matrices!
MOHAMMAD JAWISH ;)

input output matrix

Suppose that the economy of a certain region depends on three industries: service, electricity and oil production. Monitoring the operations of these three industries over a period of one year, we were able to come up with the following observations:

1. To produce 1 unit worth of service, the service industry must consume 0.3 units of its own production, 0.3 units of electricity and 0.3 units of oil to run its operations.

2. To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of service, 0.1 units of its own production, and 0.5 units of oil.

3. Finally, the oil production company requires 0.3 units of service, 0.6 units of electricity and 0.2 units of its own production to produce 1 unit of oil.

Find the production level of each of these industries in order to satisfy the external and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system.

Solution Consider the following variables:

1. p₁= production level for the service industry

2. p₂= production level for the power-generating plant (electricity)

3. p₃= production level for the oil production company

Since the model is closed, the total consumption of each industry must equal its total production. This gives the following linear system:

The input-output matrix is

and the above system can be written as (A-I)P=0. Note that this homogeneous system has infinitely many solutions (and consequently a nontrivial solution) since each column in the coefficient matrix sums to 1. The augmented matrix of this homogeneous system is

which can be reduced to

.

To solve the system, we let p₃=t (a parameter), then the general solution is

and as we mentioned above, the values of the variables in this system must be nonnegative in order for the model to make sense; in other words, t≥0. Taking t=100 for example would give the solution

Laura Geagea

matrices simplifie software programming

http://news.povray.org/povray.programming/thread//

Homogeneous matrices are indeed a purely academic invention in computer graphics courses used to show how "great" they are, while in reality they just obfuscate the subject matter to impress the unsuspecting student. Poor teachers and book authors usually need them, while in real life you implement the code to take human-readable forms of transformations (aka rotation, translation, scaling) and turn them into homogeneous matrices once. Then you never need to know about them any more as their mathematical properties are fairly simple (well, matrices in general are simple after all) - it is only visualizing homogeneous matrices content's meaning in 3D that is non-intuitive.

charles sawma

3D transformation matrix compression and decompression

Compressing and decompressing techniques for transformation matrices 3D computer graphics systems use to animate objects achieve high compression ratios by taking advantage of common characteristics of homogenous 3D transformation matrices. The techniques use a bitmap to encode information on locations of ones and zeros of the matrix--bypassing the penchant of compilers to represent such information as high-precision numbers. Since most video game processors and display hardware are constrained by their resolutions and since an original transformation matrix often stores data that is more accurate than necessary, the techniques convert some real numbers in the matrix (e.g., those within the range of -1 and 1) into integers by scaling them by a constant. The resulting compressed matrices occupy much less storage space than their non-compressed counterparts, and can be efficiently decompressed in real time for use in interactive real time 3D animations.

for bigger explanation and full example http://www.patentstorm.us/patents/6591019/description.html

Matrices and lines

There are 2 uses of matrices when it comes to lines:

1_Test for collinear points:
so lets say we have these 3 points: (x1,y1) (x2,y2) (x3,y3)
the rule is: three points are collinear if and only if the determinant found by placing the x-coordinates in the first column, the y-cooredinates in the second column , and one's in the third (that's just because we have 3 points so we need to have a square matrix so we add a column of 1's) is equal to 0
and we ask the question:is the determinant of this matrix
x1 y1 1
x2 y2 1 equal to 0? if yes then they are collinear
x3 y3 1
2_to find the equation of a line:
we take to of these 3 points and we put them in a 3*3 matrix with the first row the 2 variables x and y and the determinant of this matrix shoud be equal to 0:
x y 1
x1 y1 1 =0
x2 y2 1
let's take a simple example:
(1,1) (2,2) (3,3)
1_determinant of :
1 1 1
2 2 1
3 3 1
is equal to :2+3+6-6-3-2=0 so they are collinear
2_the equation: determinant of
x y 1
1 1 1 =0 we get the equation :x=y wich is correct
2 2 1
so this method could be useful to find out if a big number of points are collinear
you can find other uses for matrices on this website:
http://people.richland.edu/james/lecture/m116/matrices/applications.html
Peter Dagher

Real life uses of matrices

First Sorry it took so long to post
now this one may seem a little hard at first but try to solve it before looking at the solution cause it's actually very easy:
we want to obtain at the end 10 L of gasoline containing 2% additive (which means 98% gasoline)
we have these 3 types of gasoline:
1_gasoline without additive
2_gasoline with 5% additive
3_gasoline with 6% additive
And finally we need to use 4 times as much pure gasoline as 5% additive gasoline.
so how much of each type is needed?
you should get :6.4L of 1 1.6L of 2 2L of 3
Given up?
the solution is too big to post here so u can look at this website and press on "ANSWER":
http://www.intmath.com/Matrices-determinants/6_Matrices-linear-equations.php


Peter Dagher

http://www.box.net/shared/9sjm9vi6x3
Here are some problems

Business Math

A football stadium has 60,000 seats. The manager divides the stadium into two sections for the exhibition games, the games the home team plays outside its division, and the games it plays within its division. The tickets are worth $19 in section A and $14 in section B. Assuming that all tickets can be sold, how many seats must he assign to each section on says of an exhibition game, a game outside the division, and a game within the division to bring in the following revenues: $970,000 for each exhibition game, $990,000 for each game outside the division, and $1,120,000 for each game within the division.

Business Math

A small fish market sells only tuna and salmon. A tuna costs the fish market $1.75 per pound to buy an $1.00 per pound to clean and package. A salmon cost the fish market $2.25 per pound to buy and $3.50 per pound to clean and package. The fish market makes $1.30 per pound profit for each tuna it sells and $1.60 per pound profit for each salmon it sells. The fish market can spend only $200 per day to buy fish and $225 per day to clean and package the fish. What is the maximum profit the fish owner can make per day and how much of each type of fish can he buy?

Quantum Mechanics

Quantum mechanics is a description of the physical reality at the atomic level of matter (molecules and atoms) and the subatomic (electrons, protons, and even smaller particles).
And Matrices in Quantum Mechanics serve as a key tool for describing the scattering experiments which form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.
http://www.chem.tamu.edu/rgroup/hughbanks/courses/673/handouts/qm_notes.pdf
An example is included above in the link, i could have typed it, but since it is long and it is Quantum Mechanics we are talking abou here i prefered no to post it.

Walid Wehbe

finding the area of a triangle

LINK:www.pballew.net/detrmnt.doc

for example if your given a problem about finding the area of a triangle with this given information A(2,1);B(5,6);C(9,-1).If you don't know
the determinant you will have to find the lenght of AB AC and BC then find the area we get 20.5.But there is an easy way to find the area is by the method of
determinant we put the x values in the first column then the y values in the second column and in the third column we put 1.Then we do the method
of determinant and we get -41.So 41 is half 20.5.And we can get + if we interchange the rows.

first method:
AB=root 34
AC=root 65
BC=root 53
if we add them all we get 20.5

method of determinant
det{2 1 1}
{5 6 1}
{9 -1 1}
= -41
Finally if we want to find the area of a triangle all we have to do is find the absolute value of the determinant over 2, it is easy, simple
and we can save time.

Sami Nassif