Thursday, December 31, 2009

Keep Up The Good Work and Happy New Year!

Dear all,


So far, we have 7 wonderful posts!! Thanks a lot for your effort and great work! We are still waiting for the remaining students to actively post and comment!


Happy New Year for all!!


Please do not forget to:
  • Keep your post short, precise and clear
  • Keep checking the Blog on a daily basis and you are welcome to comment on more than 1 post
  • Be clear in the title of your Post: the title should explain the subject of your Post
  • Work on your profile (pictures, etc)
  • Link this Blog to other interesting math blog
  • Answer the POLLING questions at the top right corner of the Blog
For FUN, check my Blog and Wiki whose links are under Followers! I created them in Alaska while teaching online!


Good Luck,


Zeina

Roy Doumet post adress

http://www.richland.edu/james/lecture/m116/matrices/applications.html
CLICK ON MY POST TO ENLARGE IT CHECK IT IT'S INTRESTING

Wednesday, December 30, 2009

SCIENTISTS USE MATRICES AS SUPPORT TO THEIR STUDIES.

In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters. Matrices can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory.In this article, the entries of a matrix are real or complex numbers unless otherwise noted.More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of rectangular arrays," in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination.Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps!The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory.

Linear Algebra in Chemistry

Determinants and the Eigenvalue problem

In 2nd year quantum chemistry you will come across this object: \begin{vmatrix} \alpha - E &  \beta  & 0 & 0  \\ \beta & \alpha -E & \beta  &  0  \\ 0 & \beta & \alpha -E & \beta    \\ 0 & 0 & \beta & \alpha -E    \\ \end{vmatrix} = 0
You divide by β and set (α − E) / β to equal x to get:
\begin{vmatrix} x & 1  & 0 & 0  \\ 1 & x & 1  &  0  \\ 0 & 1 & x & 1    \\ 0 & 0 & 1 & x    \\ \end{vmatrix} = 0
Expand this out and factorise it into two quadratic equations to give:
(x2 + x − 1)(x2x − 1) = 0
which can be solved using x = -b \pm etc.

Simultaneous equations as linear algebra

The above determinant is a special case of simultaneous equations which occurs all the time in chemistry, physics and engineering which looks like this:
\begin{pmatrix}  ( a_{11} - \lambda ) x_1 + a_{12} x_2 + a_{13} x_3 .....  + a_{1n} x_n = 0   \\  a_{21} x_2 + ( a_{22} - \lambda ) x_1 + a_{23} x_3 .....  + a_{2n} x_n = 0   \\ .....  \\ .....  \\  a_{1n} x_n + a_{2n} x_n + a_{n3} x_3 .....  +  ( a_{11} - \lambda ) x_1 = 0\\ \end{pmatrix}
This equation in matrix form is ({\mathbf  A} - \lambda {\mathbf 1}) {\mathbf x} = 0 and the solution is {\rm Det }({\mathbf A} - \lambda {\mathbf 1}) = 0.
This is a polynomial equation like the quartic above. As you know polynomial equations have as many solutions as the highest power of x i.e. in this case n. These solutions can be degenerate for example the π orbitals in benzene are a degenerate pair because of the factorization of the x6 polynomial from the 6 Carbon-pz orbitals. In the 2nd year you may do a lab exercise where you make the benzene determinant and see that the polynomial is
(x2 − 4)(x2 − 1)(x2 − 1) = 0
from which the 6 solutions and the orbital picture are immediately obvious.
The use of matrix equations to solve arbitrarily large problems leads to a field of mathematics called linear algebra.
Alexander Abi Chaker (20091978)

MATRICES USED TO BALANCE CHEMICAL EQUATIONS

Here's the article link: http://www.shodor.org/unchem/math/matrix/index.html

When I read this article, I found it to be very useful, specially for Chemistry students interested in applying techniques of algebraic maths in their studies.
It shows how matrices can be very easy to use in order to balance even the hardest chemical reactions.
For instance, in this reaction: MgO + Fe ---> Fe2O3 + Mg (unbalanced reaction), we simply add coefficients: a, b, c and d. The new reaction: aMgO + bFe ---> cFe2O3 + dMg
For Mg: 1a + 0b +0c = 1d
For Fe: 0a + 1b - 2c = 0d
For O: 1a + 0b - 3c = 0d
We now can balance the reaction using the matrices formulas as shown in the article above.

This application was very interesting and it taught me that a matrix can be applied in almost every aspect of study in other fields such as physics, computer science, accounting and as shown here computational chemistry.
It is amazing how elementary numbers can solve so many complex issues!!

When you click on it, it opens in a larger FONT

Tuesday, December 29, 2009

writing short messages using numbers and matrices

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
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  = 
5 919 /1 10 15/ 2 4 11 
and it means ”CERNY PSIK”(a black
dog). Source:http://www.mff.cuni.cz/veda/konference/wds/contents/pdf06/WDS06_106_m8_Ulrychova.pdf
what did i learn: i chose this article because its very simple and useful.It helps me text message others in a simpler way and in a fast way.I liked it and i can use it on daily bases.

BY JOHN Michel HItti

Matrices & Cryptography

Cryptopgraphy is extremely benifical to people, enabling them to keep certain information private, it has been a very imporatant matter all throughout history and till this very day.
You Probably Wonder how can matrices possibly be used Cryptography!
Well it is quite simple to say that due to the existence of matrices their multiplication and their inverse, The idea of cryptoghraphy was made possible.
Here is an example I found demonstrating how cryptography can be use to hide certain information using matrices multiplyong them and then finding their inverses, if you are intrested enough check this link to expand your understanding on how the following example was made Possible.

EXAMPLE:
Consider the Following Matrix Reffered to as the Encoding Matrix:
{-3,-3,-4;0,1,1;4,3,4} ( the comma seperates the entries in the rows and the semicolon seperates the columns)
Let the message we want to send be: Prepare to negotiate.
Assing every letter of the alphabet to a number ex: A-1 B-2 C-3 etc...
By Decoding the Matrix above and finding its inverse relate every entery in the Inverse matrix to the approtpriate letter and you will find your message.
Check out the Link To Deepen your understanding of Cryptography using matrices and their inverses. http://aix1.uottawa.ca/~jkhoury/cryptography.htm

Thank You,
Oliver Chlela

Matrices Multiplication in Calculating a Population's New %

Here's the link for the article I found: http://www.sosmath.com/matrix/matrix1/matrix1.html
It tells how multiplying matrices is useful in life!!
They give an example about a city with inner and suburb kinds of populations and that every year, a certain % of the inner population moves to the suburbs and vice versa. In order to find the new % of the inner and suburb population, they represented the initial population of the inner city with respect to the suburbs with a 2x1 matrix, then multiplied it on the left by a 2x2 matrix representing the variables. Therefore, using matrices multiplication, they can calculate the new % of population in each part of the city for as many years as they like!! :)

So matrices are really an easy and simple way to solve some problems because, in this previous example, using another formula woul've made things hard to solve (ex: the % of the population after 1o years might get so complicated) while using matrices makes things very delicate!!


SANDRA ABOU HARB 20091661

Matrices used to solve a statics problem

"http://www.intmath.com/Matrices-determinants/6_Matrices-linear-equations.php" exercise 4 in this page is a common statics problem were we need to find the forces acting on the rod such that the rod is in equilibrium.
we get 3 equations with 3 unknowns after we sum the vertical and horizontal forces and the moment along any point on the rod.
and then we write them
In matrix form, we write the equations as:
forcesforces
using the inverse method we solve for F1, F2, and F3.

So the solution for the system is:
solutionsolution 2solution 3
So
F1 = 425.5 N
F2 = 1079.9 N
F3 = 362.2 N

Sunday, December 27, 2009

Another interesting website!

I just found the following: http://www.sosmath.com/matrix/matrix0/matrix0.html

Check it out! :)

Zeina

Wednesday, December 23, 2009

Raed's post

Dear Raed, your post was really interesting: "About How is Math used in nearly everything!", but it was TOO LONG of a post, and second, it is NOT related to Matrices and Determinants.
So, look at my post and see how small, practical and straight to the point it is, and then Look for another Post! ok?
I deleted your post but I have it as a PDF file and will send it to you via email.
Sorry about that!
Try again!
Thanks, Zeina

Thursday, December 17, 2009

READY???

DEAR ALL,
I HAVE INCLUDED AN EXAMPLE BELOW and a comment from a virtual student called Darin :)
READ IT CAREFULLY AND THEN FIND YOUR OWN POSTING AND POST IT! DEADLINE IS JANUARY 4, 2010 TO:
1)POST
2)COMMENT ON A MINIMUM OF 1 POST
GOOD LUCK!
ZEINA

An example of Real -Life Applications of Linear Algebra

Hello!

Since, as Julien mentioned, it is complex to post matrices, and other mathematical notations on the Blog, then I do recommend the following:

Once you find a nice article online, Copy and Paste the Link BUT tell us in your own words more details about your example and why do you think it is an interesting post!

Example:
My article is found at: http://www.mff.cuni.cz/veda/konference/wds/contents/pdf06/WDS06_106_m8_Ulrychova.pdf

In this article: Example 1 shows us a real life situation of a Shopping scene! 3 people who would like to buy some rolls, buns, cakes and bread from 2 shops. The question is: "which shop is the best to pay as little as possible?"
Tables are given on quantity and prices.
By rewriting each table as a matrix (first table P and second table Q), we can find, for example, the amount spent by Person 1 in Shop 1, by multiplying row 1 of P with column 1 of Q!

Nice, simple and easy!

You can get more details by reading attentively example 1 in the attachment I provided.

What did I learn???
I chose a problem that is used in a Business Math course, and shows how matrices can really be effective and quick in case we have several variables. A matrix gives order to things and facilitate computations!

What do you think?

Zeina

Tuesday, December 15, 2009

Welcome to your Blog!!!

Welcome you all and let me tell you: YOU HAVE MADE IT SO FAR! Congrats!!!

Well, as a first step, take a tour of the Blog following the instructions I gave you, and enjoy editing your profile!

Next week, I will POST the Real Assignment that you will have to do over Christmas Break!

See you,

Zeina

Tuesday, December 1, 2009

Real World problems in Linear Algebra

Dear Linear Algebra NDU students,

I was approached many times by some of you, asking me: "What is the use of solving linear equations using matrices, the inverse of matrix, determinant, etc in real life situations, in Business, in Engineering?"

Well, I think that this is a very important observation! One must see the whole picture in order to make sense of what one is studying! right?

So, I thought of starting this Blog! We can build it over time with topics and share ideas and learn from each others. But also, we can keep in touch after the semester! :)

Zeina