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April 5th, 2018, 01:20 AM   #1
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Matrix

Is there a matrix structure (the structure of rows and lines (in Hebrew, we describe it as a table) has meaning?
What by putting number in lines and rows the operation of matrices has a meaning?
If I put the numbers in another way (not as table), will it will be another structure?
And my last question is Why the operation has a meaning in matrix?

Last edited by skipjack; April 5th, 2018 at 12:08 PM.
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April 5th, 2018, 05:10 AM   #2
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Sorry, but I can't make any sense out of that. I don't know what you are asking.

A "matrix" is an array of numbers (your "table") together with specific definitions of addition and multiplication. For example, $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}+ \begin{bmatrix}p & q \\ r & s \end{bmatrix}= \begin{bmatrix}a+ p & b+ q \\ c+ r & d+ s\end{bmatrix}$ which is often called "component wise" since we add the two components in the same place in each matrix. Notice that two matrices must have the same number of rows and the same number of columns in order to be added.

To multiply two matrices, think of the rows of the first and the columns of the second as "vectors" and take the "dot product" of each row with each column. The product $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix} \begin{bmatrix}p & q \\ r & s \end{bmatrix}= \begin{bmatrix}ap+ br & aq+ bs \\ cp+ dr & cr+ ds \end{bmatrix}$.
To multiply two matrices, we only require that the number of rows of the first matrix be the same as the number of columns of the second matrix.

An important application of matrices is to write systems of equations as a single equation. The system of equations
ax+ by= c
dx+ ey= f

can be written as $\displaystyle \begin{bmatrix}a & b \\ d & e \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}c \\ f \end{bmatrix}$ and then, writing $\displaystyle A= \begin{bmatrix}a & b \\ d & e \end{bmatrix}$, $\displaystyle X= \begin{bmatrix}x \\ y \end{bmatrix}$, and $\displaystyle C= \begin{bmatrix}c \\ f\end{bmatrix}$, we can write that equation as $\displaystyle AX= C$.

I don't know what you mean by "put the number in another way (not as table)". If you have a set of numbers (not a single number) written in some other format, then it is not a "matrix". What it would be depends upon the way it is written.
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Last edited by skipjack; April 5th, 2018 at 12:10 PM.
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April 5th, 2018, 05:24 AM   #3
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I try again to ask:
The structure of matrix - what is it?
A definiton, an axiom or something else?
The definition in my books say - definition is object that defined by other basic object?
So what the definition of "matrix"? Is it an axiom? Or build on other axioms?
The definition that I describe is based on my geometry book.
Mybe my English in wrong and the term is not "definition"? [So what the term on object that depends on basic other thing [I think that called axiom - please somebody what order in the terms?]
Thanks.

Last edited by skipjack; April 5th, 2018 at 12:11 PM.
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April 5th, 2018, 06:57 AM   #4
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Math Focus: Dynamical systems, analytic function theory, numerics
A matrix is nothing more than an array of numbers and it serves to neatly organize information. Its meaning can vary widely and is context dependent. In math, the most common use is to represent a linear transformation between vector spaces with respect to a fixed basis. In this case, the action of the transformation on each basis vector is described by a single column of the matrix. Other uses are plentiful, especially in math, computer science, and physics.

Any operations performed on matrices also depend completely on the context. For instance, if two matrices represent linear transformations, then adding them has a particular meaning which is useful in the theory of vector spaces. On the other hand, if you have matrices which contain the ages of the children for the employees at your company, it is probably pretty meaningless to add them together.
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Last edited by SDK; April 5th, 2018 at 07:01 AM.
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April 5th, 2018, 07:40 AM   #5
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Thanks.
If I think on what you said - this not really a creature in mathematics.
It is only a way to arrange data, so with the "correct" arrange the operations are work.
Right?
It is not base on axiom.
I think matrix is a kind of syntax like (), [] and etc.
It is a syntax that made to help you dealing with numbers and vectors (what else - please, you tell me about else).
Right?

Last edited by skipjack; April 5th, 2018 at 12:12 PM.
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April 5th, 2018, 08:12 AM   #6
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I disagree with SDK. What he is describing is simply an array. To be a matrix it has to have multiplication and addition defined as I said above, so that a matrix is an algebraic object. policer, what did you not understand in my first post? I will try to explain anything in it you did not understand.
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April 5th, 2018, 08:15 AM   #7
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I get lost with the expression "component wise".
What is its meaning?
In simple words...

Last edited by skipjack; April 5th, 2018 at 12:13 PM.
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April 5th, 2018, 08:54 AM   #8
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Quote:
Originally Posted by policer View Post
I get lost with the expression "component wise".
What is its meaning?
In simple words...
The "components" are the individual numbers in the matrix. When I say matrix addition is "component wise", I mean that we add the two upper left numbers to get the upper left number in the sum matrix, the two number in the top row to the right of that to get the number to the right of the top left to get the number in the sum matrix to the right of the top left.

More generally, we add the number in the "ith row, jth column" in one matrix to the number in the "ith row, jth column" to get the number in the "ith row, jth column" in the sum matrix. It's pretty much what you would expect to do.
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Last edited by skipjack; April 5th, 2018 at 12:14 PM.
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April 5th, 2018, 08:58 AM   #9
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Thanks, I need to leave the computer, but when I'm back I will read your post in better way.
Thanks, again.

Last edited by skipjack; April 5th, 2018 at 12:06 PM.
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April 5th, 2018, 09:29 AM   #10
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Can you post a picture, it is hard to me to follow.
But if you don't want it o.k.
Or you the formula with your explaination to describe the metrix {If I clear use the mathematic code to show me the upper and the other thing). Or bold text in the post will help too.
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