Matrix Multiplication and Coordinate Systems
1. Matrix Multiplication
1.1 Definition
A matrix multiplication is the production of a new matrix from a pair of matrices.
Matrices can only multiply if the number of columns for the first matrix equals the number of rows for the second matrix.
For example
Multiplying AB
A ---> 3x2 matrix (3 is the # of rows, and 2 is the # of columns)
B ---> 2x3 matrix (2 is the # of rows, and 3 is the # of columns)
THEY DO CAN MULTIPLY!
The new matrix will have the rows of the first matrix and the columns of the second matrix.
For example
AB = C
A ---> "m x p"
B ---> "p x n"
Then C will be "m x n"
1.2 Dot Product
$ A= \left(\begin{array}{cccc}a1\\a2\\.\\.\\.\\an\end{array}\right) $ $ B= \left(\begin{array}{cccc}b1\\b2\\.\\.\\.\\bn\end{array}\right) $
A*B = a1b1 + a2b2 + ... + anbn
In order to make a matrix multiplication, there should be the operation of dot product between the rows of the first matrix and the columns of the second matrix. To fin the entry (a,b) in the new matrix, the sum of the products of the bth column in the second matrix and of the ath row in the first matrix.
For example
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
You take a11*b11 + a12*b21 and that's your c11.
(1)*(5) + (2)*(7) = 19
Then a11*b12 + a12*b22 = c12
(1)*(6) + (2)*(8) = 22
After no more columns, you move to the next row. Take a21*b11 + a22*b21 = c21
(3)*(5) + (4)*(7) = 43
And the last one in this case would be a21*b12 + a22*b22 = c22
(3)*(6) + (4)*(8) = 50
So, the new matrix would be
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
1.3 Matrix Multiplication in the Real World
Matrix multiplication are used in many life situations. In fact this semestre I had two clases where matrices where applied.
To solve for variables and to get probabilities are some of the main functions in the real world.
When you have equations, and need to solve for the unknown variables, a matrix multiplication and rref let you do this.
For example
Equation 1 -------> 12x + 3y = 42
Equation 2 -------> 3x + 8y = 54
Equation 3 -------> 15x + y = 36
You can put them in a matrix multiplication like this:
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
You can verify that the equations and the matrix are equal by doing the dot product.
After doing the rref, you get that the variable x= 2 and y = 6 .
1.4 Matrix Multiplication Related to Other Classes.
I have known matrices for 1 year, and used them in four classes already. Matrices as told before work for many reasons and areas. Linear Algebra, Differential Equation, Stochaistics Models, and Operational Investigation I. In many companys they use the matrix multiplication to get to how many output to produce, or how many employees to have. In Stochaistics Models the main reason of matrices where probabilities, where with a Markov Chain, you get the probability of going from one step to another.
For example
In a discrete Markov Chain of 3 steps, you have the probability of 40% to staying from step 1 to step 1, 20% from going from step 1 to step 2, and 40% of going from step 3.
Then from step 2 to step 1, it has 20%. From step 2 to stay in step 2, it has 30%. To go from step 2 to step 3 it has 50%.
Finally from step 3 to step 1, 60%, from step 3 to step 2 just 20%, and from step 3 to stay in step 3, 20%.
Where you get the matrix:
P= \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)
Excercise:
If they ask you, initially (k) it is in step 1, how many times on average does step 3 would be visit in (k+3)?
So, you need to add the entry p13 + (p13)^2 + (p13)^3. In other words, the average number of visits in k+1 would be p13, then you MULTIPLY the matrix times the same matrix and add the same entry. Again the same step but for k+3, where you take the matrix to the third.
Solution: p13 + (p13)^2 + (p13)^3 =
1.5 Worl Cup Problem
The following operation is equal to the amount of titles winned by Germany and Uruguay. Germany will be variable "x", and Uruguay will be variable "y".
Matrix Equation:
P = \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)
You do the dot product and you get that:
2x + 3y = 12
6x + 8y = 34
- Solving for this equation, you get that Germany has 3 World Championships and Uruguay just 2.
This is an easy example, but is just one of many problems that can be done with matrices multplications.
1.6 How I see It
I see matrix multiplication as something new in my life but that has a very positive use. Is not material that a middle school student knows, is something extra. But, at the same time is not that difficult. The process of solving this multiplications is with the same tools you have worked since a nine years old: multiplication, addition, substraction, etc. Don't be scared because it is a matrix operation, just step by step, get going through the problem and you should not have a problem. The only problem is to know what to do next, and what to avoid. Because there are some properties to follo in matrix multiplication, where in 1.7 would be mentioned.
1.7 Properties
2. Coordinate System
2.1 Definition
2.2 Uses in Life
2.3 Related to Other Clases
2.4 Fun example
2.5 How I see it
