Linear Algebra 2- Properties of Matrix Multiplication (2.1)

Matrix Multiplication: 
Theorem 2: Let A be an m x m matrix, and let B and C have sizes for which the indicated sums and products are defined.
a.) A(BC) = (AB)C                                                (Associative Law of Multiplication)
corresponds to composition of linear transformations.
b.) A(B+C) = AB+AC                                           (Left Distributive Law)
c.) (B+C)A = BA+CA                                           (Right Distributive Law)
d.) r(AB) = (rA)B = A(rB)
for any scalar r.
e.) I_mA = A = AI_m                                             (Identity for Matrix Multiplication)

Example: Product 

Example: Associative Law of Multiplication: 

Example: Identity for Matrix Multiplication:

Warnings: 
1. In general, AB does not equal BA.
2. The cancellation laws do not fold for matrix multiplication. that is, if AB = AC, then it is not true in general that B = C.
3. If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.

Powers of a Matrix: 
If A is an n x n matrix and if k is a positive integer, then A^k denotes the product of k copies of A.
                                                    A^k = A x A x A.....
If A is nonzero and if x is in R^n, then A^k is the result of left multiplying x by A in the amount of k times. If k = 0 then A^0x should be x itself, which will be the identity matrix.

Example: 

 











Example: 

If you still need some help with matrix multiplication here is a good link to help you out.
https://www.mathsisfun.com/algebra/matrix-multiplying.html
Also i have found a really good video on Powers of a Matrix and how to use the powers as an implication of multiply matrix's. Its called power of Matrix's and you can find it on Youtube.

An example problem could be: Find all properties from theorem 2 (a-e) and also find the power matrix's for A, B, and C.

Solving One-Variable Linear Equations

Solving One Variable Equations: 

"Linear equations are equations with a plain old variable like "x" rather than something more complicated like "x^2"

You've probably already solved linear equations; you just didn't know it. 

When solving a one variable equation, you need to "undo" whatever has been done to the variable. 

 A solution: a solution to an equation is a value or set of values that makes the equation true. 

For Example in the Equation: 

x + 2 = 3

the solution is x = 1, because 1 + 2 = 3. 

Lets Review: Additive Inverse: 

An additive number is something you add to a number to make it equal 0. 

2 + (-2) = 0

5 + -5 = 0

Multiplicative Inverse: 

A multiplicative inverse is something you multiple by another number to equal 1. 

2 X 1/2 = 1

You do this in order to get the variable by itself; in technical terms you are "isolating the variable."

For Example: x + 6 = -3

You first want to get the x on the left side by itself. So you will subtract the 6 to the right side. So now the equation looks like x = -3 - 6. So now we solve for x. -3 - 6 = -9. So x = -9.

Whatever you do to an equation,
do the   S A M E   thing
to   B O T H   sides of that equation!

Example: x - 3 = -5

Since you want to get x by itself, so you want to add 3 to both sides. 

x - 3 = -5

+3 = +3 

x = -2

Then the solution is x = -2. 

Lets solve this equation: 3x + 6 + 4x = 13

First we need to combine like terms: those terms are the 3x + 4x = 7x. 

So now the equation look like this: 7x + 6 = 13

So now to get the variable by itself we need to subtract the 6 from both sides. 

So we get 7x = 13 - 6

7x = 7

now we divide by 7 to get x by itself to both sides. 

So x = 1. 

Now we need to check our work

3 (1) + 6 + 4 (1) = 13

3 + 6 + 4 = 13

13 = 13. 

Therefore the solution is x = 1. 

Lets solve this equation: 3 - 5x = 18.

First we need to move the 3 to the other side to get x by itself, so lets subtract the 3 from both sides.

-5x = 18 - 3

-5x = 15. 

Now we need to divide by the -5 to get x by itself. 

x = 15 / -5

x = -3.

Now we need to check our work so we plug x = -3 back into the equation. 

3 - 5(-3) = 18

3 - (-15) = 18

18 = 18. 

Therefore the solution is x = -3. 

Oder of Operations

Order of Operations: 

What is the answer to this ? 4 + 6 x 2 =

Did you get 20?

Can you see how the answer could be 16?

There are two possible ways to figure out this problem. 

          Do the addition first :                                                 Do the multiplication first: 

                4 + 6 x 2 =                                                                                  4 + 6 x 2 =

                    10 x 2 =20                                                                                  4 + 12 = 16

Which is it? 

Both answers can't be right or we would always be arguing to the answers of math problems. The nice thing about math is there is always just ONE answer. 

So a long time ago mathematicians decided to make a set of rules for what to do first in a math problem. 

The rules are called the Order of Operations!

which is abbreviated by PEMDAS.

The P in PEMDAS stands for "parenthesis"

Parenthesis in math are used to group important things together, so you always do them first. 

For example: 

 Do inside the parenthesis first:

Here's another Example: 

 In this problem we have to do paranthesis first then exponents then division.

The classic sentence that math geeks have been using for years to remember the Order of Operations (PEMDAS) is: 

Please Excuse My Dear Aunt Sally

Can you think of a sentence to go along with this acronym?

Murder Mystery- Geometry Game

After a lesson on shapes, angles and measurements i will put my students to the  test with a fun interactive game called Murder Mystery. This game will get the students up out of their seats to find clues and find who the suspect is. This game is kind of like the board game clue, students have to search through clues to find the murder.

Geometry Game- Murder Mystery: 

The Scene: The police are called to a high school, lying on the floor in the library is a dead student. As police search the library they find 5 clues written down by witness. They have sent the clues to you to decipher. They also provide a list of those present at the time of the murder. (This list is all the students in the class,  with specific details on each of the students). There are 32 suspects, each clue will eliminate half the number of suspects remaining. When all clues have been solved the suspected person will be revealed. 

Clue One: Get into Shape:

The answer to each number is a letter A=1, B=2, etc. 

1. Total number of sides of all four squares.

2. An octagon has ? sides. 

3. A square based pyramid has ? vertices. 

4. A pentagon has ? sides. 

5. A parallelogram has ? sides.

There are many more questions about 20 but I'm just giving you an example of questions you can ask. 

Clue Two: Look From a New Angle: 

I will provide the students with different kinds of angles and they will have to figure out the angle measure of each line. By the same system as above a = 10 degree, b = 20 degrees, and so on. 

For example: 

The rest of the clues also have to do with things the students learned from this particular lesson, they will have a clue on compass/directions, and also coordinating points on a plane. The last clue i will give them a story and they will have to find the error that is wrong in that story to find the suspect. 

I got this idea from another teacher but put my own spin on it, I will use my actually students as the suspects and I will let them choose if they wanna work together or not. None of the students will know who the suspect is until the game is over. This gets them up and moving around and also has them use their ipad's for clue number two. The story I also provide them with will also be available on their ipad's so they can see if they can spot what doesn't belong in the story. I have never done this game before but am looking forward to trying it in my classroom one day. I think this game can be for any grade and for any subject. The teacher just needs to have fun creating the questions, and the students need to have fun trying to figure out the clues. 

Welcome to my Technology Mathematical Educational Blog

This is a blog that I have already created to enhance students knowledge about specific topics in Discrete Mathematics (also know as Calc 4). I will be incorporating technology into my classroom also so many of my posts will be ways to incorporate technology into the mathematical classroom. I will be sharing different lesson plans, activities and my thoughts on certain math topics. Can't wait to hear feedback from my fellow classmates about incorporating technology into the classroom. At the rate technology keeps improving, we as teachers need to find new ways to keep up with the students and make learning more interesting to them. 

Graph Coloring

Graph Coloring: 

In Graph Theory, graph coloring is a special case of graph labeling. It is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In the easiest form it is a way of coloring the vertices of a graph so that no two adjacent vertices share the same color, this is called vertex coloring. An edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. 

A proper vertex coloring of the Petersen graph with three colors

The convention of using colors originates from coloring the countries of a map, where each face is literally covered. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. For example: In mathematical and computer representations, it is typical to use the first few positive or nonnegative integers as the colors. In general we can use any finite set as the "color set." Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the public in the form of the popular number game Sudoku. 

There are twelve different ways to color the map from above using three colors. 

A coloring using at most k colors is called k-coloring. The smallest number of colors needed to color a graph G is called its chromatic number, and is often denoted X(G) or Y(G). The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color. 

Examples: 

Examine Questions: Find the chromatic number for each graph listed below?  

Citations: Wikipedia, and Google images

Stable Marriages

The stable marriage problem is another classic appreciation of discrete mathematics to social networks. 

Suppose that you have been hired to be a matchmaker with n men and n woman who wish to be paired up with their perfect matches. Each man provides the matchmaker with a list of the woman ranking from first choice to last choice, as same for the woman. 

The matchmakers job is to provide a way to pair up the n men and n women in such a way so no man or woman is left behind. 

The stable marriage theorem states that your job can always be accomplished, no matter how the men and women have ranked one another. This theorem provides an algorithm that accomplishes your task. 

In round 1: every man proposes to his first choice. those woman who receives the offers tentatively accept the best offer and tell the other men not to come back. The rejected men then propose to their next choice, and the women then tentatively accept the best offer they have so far received and tell the other proposers to go away. This process continues until every man and woman are paired up. 

Example One: Suppose the matchmaker receives the following list: 

man 1 (3,1,4,2,5)

man 2 (1,3,5,2,4) 

man 3 (5,4,3,2,1)

man 4 (1,5,4,2,3)

man 5 (3.4.5.1.2)

woman 1 (3,5,1,4,2) 

woman 2 (3,1,2,4,5)

woman 3 (1,2,3,4,5)

woman 4 (5,3,1,2,4)

woman 5 (5,4,3,2,1)

 Use the stable marriage algorithm to find a stable parring. 

Answer: Men 1,2,3,4 and 5 will propose to women 3,1,5, 1, and 3 respectively. Woman 1 rejects man 2, woman 3 rejects man 5. So then men 2 and 5 propose to women 3 and 4 respectively, but man 2 is rejected again because woman 3 is already taken. Man 2 now continues to move down his list to woman number 5, who accepts him and rejects man number 3. Now man 3 proposes to woman 3, who rejects him. Then man 3 proposes to woman 2 who accepts his offer. WIth no man or woman unattached, the algorithm terminates with stable pairings: 

man 1 with woman 3, man 2 with woman 5, man 3 with woman 2, man 4 with woman 1, and man 5 with woman 4. 

Examine Question: Suppose the matchmaker receives the following list: 

man 1 (c, a, d, b, e)

man 2 (a, c, e, b, d) 

man 3 (e, d, c, b, a) 

man 4 (a, e, d, c, b)

man 5 (c, d, e, b, a)

woman a (3,1,2,4,5)

woman b (3, 5, 1, 4, 2)

woman c (5,3,1,2,4)

woman d (1,2,3,4,5)

woman e (5,4,3,2,1)

A. Use the stable marriage algorithm to find a solution pairing. (men proposes to women)

B. Find another stable marriage obtained by having the women propose to the men.