Probability : randBin

Hi everyone! We didn't really learn anything new today. It was basically just
a review of the pasts lessons and homeworks. Today's post will be about randBin.
Most people are having troubles understanding how randBin works, including
myself, but I tried to do a research and learn more about it.

randBin(
meaning "random Binomial", generates a random integer between 0 and n from a Binomial distribution.

How to use the randBin command:

1. Press [MATH]
2. Scroll to the farthest right until you get to PRB and press enter.
   
3. Select and press [randBin(]
   
4. Enter the numbers using the formula below

randBin( n,p,# of simulations)
n = number of trials
p = probability of success


* the seed that is on rand affects the output of randBin(


Example:
Design an experiment to determine the probability of getting 50% correct answers on a 12 multiple choice questions quiz if you try to guess all the answers.

randBin( 12, 1/4, 5)

[ 2 1 4 1 2 ]
[ 1 5 1 2 1 ]
[ 4 2 2 5 0 ]
[ 2 4 3 4 5 ]
[ 3 3 1 4 2 ]

Answer : P ( 50% or over ) : 3\ 25= 12%



and that's what i understand about the randBin( command. Please feel free to correct any mistakes or add in some information(s) that is/are missing.




next up is ... Eugene.

REMINDER

Make sure to click the spell check button before you hit the big friendly orange PUBLISH button. You can delete this now.

Slides February 26th

Here they are ...

Probability04

View more documents from heviatar. (tags: randbin trees)

Scribe Post, Pascal's Triangle

The news for today is: *drum rolls*
Hey Hey guys~! In today's class we began by looking at Pascal's triangle. First off we all looked at Pascal's triangle as Dr.Eviator began encouraging us to take a step up and find the next row, which the triangle looks like this and the red numbering is the next row is the answer, here is the picture so you guys can see what we did.

Pascal's triangle is a geometric arrangement of binomial coefficient's which is one of the reason why this is mentioned in today's class.
We start the first row with a 1 because 0+1 = 1

Hence is why we the result in the second row is two 1's. Then it continues the pattern which is adding the top number's which for the 3rd row you would add 1+1=2. For the 4th row to get the 3's you will add 1+2=3. For the 5th row you will add 1+3=4, 3+3=6, 3+1=4 and the one's at both ends.
We also looked for patterns in the triangle and there were a couple that everyone can see just by looking at it. I also found an animation on the internet which is found in at this url. http://upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif


I hope the animation I found helps you come to an understanding of how the triangle works a bit more visually. In another sense here is also another way you can look at Pascal's triangle. It is used in today's word problem. Which basically first we had to find how many ways we would pass the post office which was 3 ways. The next question was how many ways we can find to go the school which was 35 ways. And finally we had to find the probability of passing the post office on the way to school. Well that's all for today ladies and gentlemen sorry for the late post I was exhausted today and took a bit of a long nap>.< Lamael~!

Slides February 25th

The Internet is back, yay! Here they are:

Probability03

View more documents from heviatar. (tags: pascal probability)

Experimental/ Theoretical Probability

Today, we started our class by doing a group problem. We were ask to experiment, play and take risks for the problem. The problem was entitled "A Great Way to Make Money". It was similar to the activity that the class did yesterday. After, we went over it in class and figured it out together.(everything we did should be in the slide)

We didnt learn anything new today, we just pretty much reviewed everything that we learned yesterday.

***Things that we need to always remember to be able to solve the probabilities.
1. Experimental probability --> is the chance of an event happening as determined by calculating the mathematical result.

........................number of times an event occurs
probability = ___________________________

........................total number of trials

2. Theoretical probability --> is the chance of an event happening based in repeated testing or observation

.......................number of ways the event can occur
probability = ____________________________

.......................total number of equally likely outcomes

The class finished off by solving the homework from yesterday (everything should be in the slide too) .

REMEMBER TO KEEP REVIEWING BECAUSE WE WILL BE HAVING A QUIZ SOMETIME THIS WEEK!

that'd be all everyone! :)

The next scribe is DAVID.

Slides February 23rd

My apologies for the delay. Here they are ...

Slides February 23rd

View more presentations from heviatar.

Theoretical Probability Feb23/09 , Dr Eviatar

Well i was posted scribe for Friday but that was our test day.

So...
Today Dr. Eviatar went over last class on probability mentioning the questions on pennies. The Internet was done today's class so she couldn't show us a website she had planned but we got work done anyway!

Today we looked at Theoretical probability

Difference between Theoretical Probability and Experimental Probability

Experimental Probability is when a experiment is needed to show the Probability of something occurring. Such as how many times you will get head on a coin after 25 flips.

Theoretical Probability is a stated fact. Such as having a 50 percent change of either having a boy or a girl.

Our main question looked at was..

In a family of 3 children what is the probability that tow of the children will be girls?

It is a fact that you can have hundreds of children of a certain gender and it will not effect the probability of getting a 50 % (1/2) chance of getting a boy or a girl next.

BBB
BBG
BGB
BGG
GBG
GGB
GBB
BGG



The red text is the only three outcomes of the
diagram at left that shows two girls out of three
children


There are eight outcomes and only three have tow girls out of three children.

So the probability of having two girls in a family with three children is 3/8

We also learned about
Binomial experiments.

Binomial experiments are experiments that has only two possibilities such as flipping a coin there is only

1. heads
2. tails

Towards the end of class Dr Eviatar allowed us to do questions from the homework, and gone over it.

okay that is all that went on in class today! next scribe is!!!!!.......Carmel!

p.s. sorry for the late post i just got off work!

Thursday, February 19th, 2009

Well Today...
We started a new unit called Probability.

I'm sure you all know the basic idea behind the word probability, but I'm going to write down a few definitions that we learned today.


* Probability: A measure of the likely hood of an occurrence of an event.

* Sample Space: A list of all possible outcomes.

* Event: A particular outcome within the sample space.

* Simple Event: A result of an experimental probability trial that is carried out in one step.

* Compound Event:A Result of an experimental probability trial that is carried out in more than one step.

* Certain events: an event that has a probability of 1 or 100%

* Impossible Events: As you can guess, it is an event which has a Probability of 0 or 0%.

IMPORTANT : Probability is shown in between 1 and 0. But also remember that probability can be relayed as a decimal, percent or fraction.

If you glance through the slides posted up earlier today, you will see the kinds of things we where looking at. The slides also explain the difference between the two different types of probability:

Theoretical Probability which is the likely hood that an event, or desired outcome, will occur.

Experimental probability, on the other hand is probability defined by DOING something.


An example of that is within the slides that depict what we did today in class. We actually tossed a set of 3 coins 20 times to determine the probability of getting a head to tail ratio of 0:3,1:2,2:1 and 3:0. Because we actually DID the experiment, it makes it experimental probability.
The formulas for finding the two types of probability above are within the slides.


That is about all I have to say about the days events.

DON'T FORGET ABOUT THE TEST ON FRIDAY EVERYONE!!!

The Next Scribe Will Be Chelsea... hope your Internet is up and running again :P

Slides February 19th

Here they are ....

AM40SFeb19

View more presentations from heviatar. (tags: experimental probability)

Today's class was our "pre-test" for our Matrices unit. This pre-test prepares us for our up comming unit test which will take place on Friday.

We were given about half the time in class to write the pre-test. After that, Mr. K split us up into groups and following that we were to compare our answers with our group members. After comparing our answers we were to hand in one pre-test for our whole group which will be worth marks. The paper each group chose to hand in was the modified test paper (meaning the answers were looked over, and corrected by what the entire group felt was the right answer within reason.)

After that, Mr. K went over every single question and explained each question step by step.

Here is an example of a similar problem which was on the pre-test.

At a local grocery store, there are 2 available sizes in which milk can be purchased in. There is the 2 litre carton, and there is a 4 litre jug. Research shows that 23% of the 2 litre carton buyers switch to the 4 litre jug on their next purchase. And 41% of the 4 litre jug buyers switch the the 2 litre carton on their next purchase. The original market share is 40% of the 2 litre carton and 60% of the 4 litre jug.

a.) What is the market share for each the 2 litre and the 4 litre in the next round of purchases?

Well in order to find the next market share for the next round of purchases, you must write the initial state matrix, and you must also write the transition matix. After that, you must multiply the two together inorder to firgure out the market share for the next round of purchases.

The inital state matrix is the original percentage market share, which is 40% for the 2 litre and 60% for the 4 litre. Also, you must remember that the percentages in a state matrix should always be expressed as a decimal.

Let's call the initial state matrix " I "


Now we must figure out the transition matrix.

The information given states that 23% of the 2 litre carton buyers switch to the 4 litre jug on their next purchase. And 41% of the 4 litre jug buyers switch the the 2 litre carton on their next purchase.

So since 23% of the 2 litre buyers switch to the 4 litre, 77% remain as a 2 litre buyer. And since 41% of the 4 litre buyers switch the the 2 litre, 59% remain as a 4 litre buyer.

Lets call our transition matrix " T "



Now we must multiply the two matrices together (I*T) in order to get the market share for the next round of purchases. Lets call the resultant matrix "R"


Now the resultant is..

Meaning the market share between the two is 55% for the 2 litre carton and 45 % for the 4 litre jug.

b.) In the long term, what would the market share be for both sizes?

Well, in order to find the long term market share you must chose a large amount of years. Lets say 50. And then you must multiply I by T to the power of 50.

So it should look like this.
L is the resultant.


Now, we must put up the power by 1 making it 51 to clarify that the percentages have stabilized.

So we now find that the matrice has stabilized. Making the final market share 64% for the 2 litre carton and 36% for the 4 litre jug.



*JUST A REMINDER! *



The final unit matrices test is on Friday, Febuary 20. Remember to study and to be prepared!



I choose K_Hannah to scribe next:)!

The Scribe List

This is The Scribe List. Every possible scribe in our class is listed here. This list will be updated every day. If you see someone's name crossed off on this list then you CANNOT choose them as the scribe for the next class.

This post can be quickly accessed from the [Links] list over there on the right hand sidebar. Check here before you choose a scribe for tomorrow's class when it is your turn to do so.

IMPORTANT: Make sure you label all your Scribe Posts properly (Your Name, Unit Title, Scribe Post) or they will not be counted.

Cycle 3

MAC Camilla chelsea Niwatori-san don

Kayla jason Eugene-1 K_Hannah Lamael

henson Alvina; iris kyle Roe Daniel Amanda

Today's Slides and Homework: February 18

Here they are ...

Applied 40S February 18, 2009

View more presentations from dkuropatwa. (tags: matrices math)

Today in class we did a matrices workshop. We touched up on many things in class today like for example solving for "X" and "Y", how connections can relate and create a connectivity matrix and how putting a connectivity matrix to the power of say 3 or 4 will result in different connections. After refreshing or memory we put or knowledge into some "real life situation" word problems. As of such since we didn't learn anything new I will just post examples of the problems we did today.

Example One

Solve for "X" and "Y"

So the first step would have to be finding X, remember row by column

(So what I did here was I knew "X" multiplied by 10 equals 50.

So I divided 10 from both sides to get "X" all by itself.

50 divided by 10 equals 5 so "X" equals 5)

Next we do the same for Y

(Now knowing that "X" is 5 we use that instead.

Then following the same steps we come up with 7.)

There you have it that's how you solve for "X" and "Y"


Example Two


Connectivity matrices


So an airline called purple lines just opened and this is how there flight patterns work on a one way trip









Create a connectivity matrix











Now for a word problem


Iris is lost and can't help herself, so henson being the bright guy he is ask whats the problem.
Iris asks how can I get from Vancouver to Montreal in a two hop flight route.


So how can henson help her? Easy





( By taking the matricies and putting it to the power of 2 you get the two hop flight routes, but what your really intrested in finding is the route from V to M)



After punching this into the calculator the two hop flight from Vancouver to Montreal is 1

And so these two things are what we reviewed in class today example one shows how you can solve for X and Y and example two shows how connectivity matrices can be drawn, read from a drawing and demonstrates how you can find different connections through the use of powers. So the next scribe person I choose is....


Iris

Today's Slides and Homework: February 17

Here they are ...

Applied 40S February 17, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

MATRICES

Today's class we started off with naming countries, after we all named a country we did a little quiz about "Connectivity Matrices". After we finished correcting the quiz, he gave us a problem and we had to do it with our group and to be hand in before the end of class. There was really nothing new, test is coming up soon, just read through the previous posts and it'll help you with your studying to the unit of matrices.

Here are some examples



What is the size of the matrix?

It would be 3x4 because it has 3 rows and 4 columns

What is entry (2,3) of the matrix?

We are looking for the entry in the second row and the third column of A, which is 4

What is A + B?

Let A = and B =

Matrix addition is defined by adding the corresponding entries of the two matrices.

to get answer go 2nd matrix > hit the right arrow twice > edit > 2x2 plug in [A]

same thing for [B] then , [A] + [B] =

What is the matrix 7Z if Z =

When multiplying a matrix by a scalar, each entry in the matrix is multiplied by the scalar.

the answer is

Let A = and B = Which of the following matrices is equal to the matrix AB?

Since A is a 4 × 3 matrix and B is a 3 × 2 matrix the product AB is a 4 × 2 matrix.

the answer is


Unit test coming up soon so just continue reading the posts everyday.

... and before i end this , the next scribe is ....

HENSON

Today's Slides: February 13

Here they are ...

Applied 40S February 13, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Today in class, Mr.K put us into groups to go over our homework. Our homework was on Transition Matrices. We went over the homework and did some more review questions on Transitions Matrices.

Mr. K gave us an example of a Transition Matrix.

Example

Suppose that the population of a small island is classified into three distinct groups, children, teenagers and adults and that each year :

• Children are born at a portion of 6% of the adult population , 1% of the children die



• 10% of the children become teenagers , 5% of the teenagers die



• 14% of the teenagers become adults , 7% of the adults die

This year, Youngville's population is 25 000. 5000 children, 18 000 teenagers and 2000 adults.

A) Write a row matrix that currently represents the population of Youngville.

The question is asking to right down the state matrix, a state matrix shows the information that that is currently given as a constant. Which is the population of Youngsville. We would label the matrix "S" which tells us that it is the state matrix.

A = Adults

T = Teenagers

C = Children

We would usually write the matrix in alphabetical order but in this case we would write it going oldest to youngest showing the population of the city more clearly.

B) Write a 3 x 3 transition matrix that shows how the population is changing over time.

To answer this question we would need to find out how the population changes.

The following matrix is labeled "T" which stands for "transition matrix."

Confusing? Well hopefully my explanation works. Adults can't turn back into teenagers so that's why there is a zero in the A,T position. 6% of the portion of adults become kids because adults make kids, and 1% of the children die, leaving 93% of the adults stay adults.

6% + 1% = 7%

100% - 7% = 93%

Teenagers can't go back into children so T,C becomes zero. 14% become adults, and 5% of the teenagers die, leaving 81% of the teenagers still teenagers.

14% + 5% = 19%

100% - 19% = 81%

Children can't just be adults right away so C,A becomes zero. 10% of the children become teenagers and 1% of the children die. Leaving 89% of the children stay children.

10% + 1% = 11%

100% - 11% = 89%

So that's how I understood the example Mr.K gave us. Hopefully this helped people understand it more.

IMPORTANT NOTE: Mr. K and the class decided that tomorrow will be another review class and that the pretest was moved to Tuesday as there is no school on Monday, Louis Riel day. The unit test will take place on the Friday of that week which is the 20th of February. Wednesday and Thursday we will be starting on our new unit.

The next scriber I choose is....

Mac

Today's Slides and Homework: February 12

Here they are ...

Applied 40S February 12, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Transition Matrices (cont'd)

Well today in class, we spent most of our time reviewing the homework we had last night. And Mr. K just reviewed a little more about Transition Matrices, there was really nothing new for today.

Transition Matrices just shows you the probability of 2 or more things, if it would sell or if it will increase in days, weeks, or even years. Basically just probability of 2 or more things in the future.

I'll give you guys just another example of transtion matrices just in case some people are still not following. And hope this helps :).

Lets say you wanted to find out which air freshener would people use more in 10 years, Air Wick or Febreze.

First you would want to write the State Matrix, which means what is happening right now. So if the 20 people used Air Wick and 35 people used Febreze then the State Matrix would be as so. A for Air Wick and F for Febreze.








Here is the Transition Diagram, you can see how people change from one thing to another or if they just stay with that one product.










So, by looking at this diagram, 90% of the people that use Febreze stay with Febreze, but 10% change to Air Wick. And 70% of the people that use Air Wick stay with Air Wick, but 30% change to Febreze.

Now you have enough information to write a Transition Matrix. Here is how it will look like.










Now all you have to do is plug in the numbers, and follow the diagram above. And it will simply look like this.










And the only thing left to do is, multiply the State Matrix with the Transition Matrix. It will look like this.








You can punch this in your graphing calculator and put the State Matrix in matrix a, and the Transition Matrix in matrix b. Then just simply multiply matrix a and b, if you did it correctly you will get an answer like this.










Now you`ve got everything, you have to find out if Air Wick or Febreze was used more after 10 years. To do that, all you have to do is take your State Matrix (S) and your Transition Matrix (T) and just multiply to the power of 10. It will look like this.










So if you have punched in the matrix correctly you should have an answer like this.







If you are having trouble with transition matrices, you can also look at the nice example Alvina gave just under this post.

*Unit test coming up soon so hope you guys have been reading the posts everyday.

annnnnnnnd last but not least...

The next scribe is...
DON !

Today's Slides and Homework: February 11

Here they are ...

Applied 40S February 11, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Transition Matrices

Today's class we started off with reviewing yesterdays homework before we moved on to 'Transition Matrices'

From what I understand, a Transition Matrices is a matrix that shows things that change in time.

Your trying to find the probability of things that you want to know in the future.

When doing a transition matrix you would have to find the state matrix ( which tells you the current state of affairs) 

and the transition matrix ( which tells you what's going to change). 

When writing a 'State matrix' you would have to know the current event that is taking place to fill in the matrices.

For example the state matrix would look like this,

B R

S = [ number of blue shoes number of red shoes]

If the person had 10 blue shoes and 2 red shoes then your state matrix would be..

B R

S = [ 10 2 ]

If you wanted to find out if a person wanted to use blue shoes, they have a 90% chance that they would use red shoes.

If the person wanted to wear red shoes, they would have a 50% chance that they would wear blue shoes.

To figure this out, you can also use a 'transition diagram' to help you 

through the process.

This way you can also see what the changes are.

So your transition diagram would look like the picture below.

 From here you can see what you need to make a 'Transition matrix' .

Then you would have to plug in the numbers. 

You transition matrix would equal to..

T = [ 6      9.2]

And that's how I believe you do a transition matrix.

The nest scriber I choose is ...

KYLE!! :)

Today's Slides and Homework: February 10

Here they are ...

Applied 40S February 10, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Today's Slides and Homework: February 9

Here they are ...

Applied 40S February 9, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Today's class Mr. K took the sheet that we did last friday and he handed our quiz back.

We then went over the homework from the slides he told us to do last Friday with the groups that he had assigned. We pretty much just reviewed everything we've done over the past week.

Mr. K gave us an example of matrix multiplication solving x and y.
Solve for x and y:


To solve x you have to take the first row and multiply it to the 2nd column.
0(3) + 8(x)=
0 + 8x = 40
x= 5
To solve y you have to take the 2nd row and multiply it to the 1st column.
-11(3) + -9(18) = y
-33 + -162 = y
-195 = y

After we solved x and y he gave us a problem and we had to do it with our group and it was to be hand in before the class ended. So if you miss today's class your lucky because we didn't do that much !

The next scriber would be Alvina ! :)

On Friday we started with a quiz on matrices. Then we went over a bit of matrix multiplication before we took a step forward onto 'Connectivity Matrices.'

Matrices can be used to summarize the routes between cities and to even calculate the different number of routes. A connectivity matrix about flight plans is a list of locations and how many ways they can be directly connected. Below is a network that shows a route service between A, B, C, D, and E...
Here is the matrix that represents the routes between them...

Our connectivity matrix must be a five by five grid because there are five 'places' A, B, C, D, & E.

For each cell in the matrix, you write a 1 to indicate that there is one route between the place in the row and column for that cell. There is one route between A to B, so there should be a 1 in the cell on the first row, second column. There is no route from A to E or from A back to A, therefore there should be a 0.
The matrix can also be written in its briefest form...



I choose Katherine to be the next scriber.

Today's Slides and Homework: February 7

Here they are ...

Applied 40S February 6, 2009

View more presentations from dkuropatwa. (tags: math matrices)

Here are the information sheets we were working with in class today ...


Connectivity Matricies

Publish at Scribd or explore others: How-To Guides & DIY connectivity matrice am40s

Niwatori-san's Corner: "welcome to the matrix"

O hayao gozimasu (that's hello formally in japanese)



Niwatori-san here again telling you the viewers what we did in class today!


Well in today's class we briefly talked about the "Digital Ethics" pros and cons about "posting online" All the digital ethics really is saying is to be precautious about your internet postings online. One little thing you don't want to say/mean online could bite you back in your future days.





And a note from Mr . Kuros saying about a John J. Medina and how exercise is the best stimulant to toggle the brain cells to work efficiently. Dr Medina says that our blood flow is rushed after doing exercise opening up the brain stimuli to take in information.



Soooo...... GET SOME EXERCISE ITZ GOOD FOR YA!!



After all the "Oath" pledging in class to follow the rules of the online discretion concerning posting. Most of us said, "Yes" we agree to the terms agreed and although we didn't get to write a signed approval Mr . Kuro's Firm grip sure is enough >___<>'_')> (kirby dancing by the way)



So remember Scalar and Vector Matrix multiplication?

If you don't here's a reminder



Scalar is the multiplication with no rules to multiplying you just multiply with the given variable/number to the given matrix

eg.

6* [ 3 2 1 = [ 18 12 6

4 2 3] 24 12 18]

this method is done by multiplying by 6 variable by each

individual number in the matrix such of 6 * 1,1 = 3

6*3 = 18 see?


Vector has rules you got to know and follow in order to get the right results.



This diagram shows you step by step "How to do?" Vector guide

EH??! You're saying my diagram looks like a paperweight newspaper?!

(well itz true it does look like it )

Anywho that is about sums (Lol SUMS that's what math does to you when start speaking in math terms) up the class from today. Oh yeah, there are some questions left to do on the blog itself. They're on the slides so check there while looking thru check out Mr . Kuros videos he posted along with the slides of today.

And to all of the people who didn't sign for blogger now is the time to do so to get easy marks people!! 10% cmon! Also read Digital Ethics over if you haven't done so it's important to know cuz you never know ya know?!

Well Niwatori-san is tired now and time is of an essense.

GAMBETE MINA!!! (work hard people!! in japanese of course!)

Next scriber is .... *drum roll plz....* (same tumbleweed passing by) "dang it all!"

*******Camilla********** YAY! *passes baton "clap clap clap" skoots off "pwoosh pwoosh pwoosh"

Today's Slides: February 5

Here they are ...

Applied Math 40S February 5, 2009

View more presentations from dkuropatwa. (tags: matrices math)

Multiplication with Matrices

There are two types of multiplication for matrices, Scalar and Vector. Scalar is the distance of an object; whilst Vector is the distance and the direction of an object. Today's notes will be focusing on Scalar Multiplication. Scalar multiplication consists of a matrix multiplied by a given number.

Examples:
1.





2.








To identify an unknown value of an matrix equation; use the existing values to find the unknown. Though said values can only be used if they have the same address.

Example:








Until tomorrow, goodbye.
As a reminder to anyone who hasn't read/watched the Digital Ethnics post, please do so now.

Today's Slides: February 4

Here they are ...

Applied Math 40S February 4, 2009

View more presentations from dkuropatwa. (tags: am40sw09 math)

Matrices Introduction: February 3rd, 2009

Let's start with a definition. "Matrix - (mathematics) a rectangular array of quantities or expressions set out by rows and columns, treat as a single element and manipulated according to rules." (courtesy of "thefreedictionary.com").

To elaborate, a matrix is always either rectangular or square, has a title (usually a single letter), and its name is derived by the title followed by the number of rows, followed by the numbers of columns.

For example:

A [1 3 -5 9] is named A, 1x4. The title of the matrix is "A", while it has one row, and four columns.

Similarly, each number (element) in the matrix is assigned an "address". The address is the lowercase letter of the matrix title, followed by the position in the rows, followed by position in the columns.

For example:

F [4 7 3 -11] is composed of:

f1,1... (4)

f1,2... (7)

f1,3... (3) and

f1,4... (-11)

Here is another matrix, with numbered columns and rows.

A 1 2 3

1[ -11 66 43]

2[ 4 74 22]

3[ 5 9 18]

4[ -8 12 -1 ]

5[ 15 -9 15]

This matrix is a 5x3 matrix, and as an example, the element found at "a 4,2" is 12.

In addition to these conventions, there are rules for matrix addition and subtraction.

Rule 1: The matrices must have the same dimensions. (such as being 1x4 + 1x4, or 8x2 + 8x2, or what have you.)

Rule 2: Matrices are added by simple addition (as well as subtracted through simple subtraction) of the respective elements on each matrix. Allow me to explain.

A [4 5 -1 -19] + T[-14 7 -3 8] is added thusly...

a1,1 + t1,1

a1,2 + t1,2

a1,3 + t1,3

a1,4 + t1,4

And the resultant matrix would appear [-10 12 -4 -11], and its title would be an arbitrarily assigned number or symbol, in context, hopefully one with some meaning.

Furthermore, there is an additional type of matrix I would like to explain. The Identity Matrix. The identity matrix is composed entirely of zeros and ones, because any number add or subtract zero is itself, while any number multiplied or divided by one is itself. Also, the identity matrix has only ones along a diagonal, with only zeros on either side. It may look something like this.

[1 0 0 0]

[0 1 0 0]

[0 0 1 0]

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In closing, everyone please check the Digital Ethics post so that tomorrow in class we can take our oaths quickly and efficiently. Also, Glenn (roe on the contributor list) will be tomorrow's scribe.