Giving a Logical "Real World" Meaning to Matrices

My main question is how to “interpret” or make a conclusion about the real world using matrices. This came from the DQ problem about two jobs that pay a fixed weekly base pay and additional pay per hour worked. The goals were:

  • To represent the equations describing the relationship of weekly base pay, hours worked and total pay.
  • To apply the rules of equations and matrices to transform the numbers into a form that is both mathematically valid and that can be used to make logical conclusions.

Summary of the Courses’ Problem

The main question were:

  1. Two jobs are represented as equations where y = total pay, x = hours worked and the constant c = base pay.
  2. This is the relationship in mathematical terms: (hours_worked * x) + base_pay = y. The equations representing two jobs are:
  • 30x + 1000 = y
  • 50x + 100 = y
    This means two jobs: one where base pay was 1000 and hourly pay was 30; another where base pay was 100 but hourly pay was 50.

The question was “How many hours for would it take for us to earn equal pay at both jobs or more for the job with higher hourly pay?”. To solve it by elimination, substitution was used according to logic: y of one job must equal the other. Thus the substitution 1000 + 30x = 100 + 50x. Since x meant the same for both equations, it was “fused” from either side and its coefficients removed to get the answer.

Question: What is the logical meaning of a matrix?

The next solution represented the two equations as a linear system:

matrix_one = np.asarray([
    [30, -1, -500],
    [50, -1, -100]  
], dtype=np.float32)

The matrix had to be solved to the point where it had the form:

matrix_one = np.asarray([
    [1, 0, 45],
    [0, 1, 2350]  
], dtype=np.float32)

This was interpreted as representing a system of equations where:

  • (1 * x) + (0 * y) = 45 -> x = 45
  • (0 * x) + (1 * y) = 2350 -> y = 2350

Rephrasing of the Question:
Does proving or placing equations into a system mean that when the value of one equation is x1, then it follows that the value of the other equations is x2 and so on?

I meant that when two equations in a transformed matrix are manipulated in such a way that the first equations x = 0 and the second equation’s y = 9, does that mean that: a system of linear equations describes equations where when equation 1’s x is 0, equation two’s y must be 9.

I concluded this because when the first equation meant x = 45 and the next equation meant y = 2350, the course went on to state that the solution was found, i.e.

  • When on job’s hours worked was 45, the total pay for the second job was 2350.
  • The value of total pay when both jobs pay equally was 2350.

In other words, my question is:
Is there any definition or formal proof for concluding that because

    [1, 0, 45],
    [0, 1, 2350] 

is a valid linear equation system, it can be interpreted as “When the first equation’s x is 45, the second equation’s y is 2350”?

Early Results of Research

1. MathInsight: About Matrices and Linear Transformations
It appears that there can be many valid interpretations of systems of linear equations:
This website gives a list of many ways in which a system of equations can be interpreted.

Math Insight: Matrices and linear transformations

Math Insight: Function Notation

I could not understand the notation well. The best I could gather was that a matrix could represent a function f(x)=Axf(x)=Ax, which has three variables (x,y,z) and outputs a two-dimensional vector (x−z,3x+y+2z)(x−z,3x+y+2z). I assume a 2-D vector usually means coordinates on a plane (x, y).

It also says this:


I interpreted this as meaning that a function g with a range of real number inputs denoted by the columns n outputs a set of n-dimensional vectors x as denoted by the rows. For this case, it seems that the input to g(x) must be a function as well, like y = (y1, y2).

2.Stack Exchange: How to Interpret Matrices
The summary I can give of this post is that the respondents see matrices as having multiple valid meanings including:

  • denoting several equations coefficients.
  • several pairs of points in a linear equation.

StackExchange: How to Interpret Matrices?

What interpretation is appropriate for this DQ course? Am I looking at the right resources for learning or are there links to better sources than these specific web pages?

Summary

Based on the DQ course I interpret a system of linear equations - which a matrix represents - as a system wherein the values of one equation inherently affect the other equations. The way in which one equation affects the other depends on two things:
1. The valid mathematical operations done on the system.
2. The meaning of the variables as defined by the problem/context, i.e. what real-life concept the equations describe.

Where am I wrong, where am I right, where should I search more for “elementary proofs”?

2020-05-28
New clarification about the answer.

Why exactly must be the total pay earned be the same? Is that one of the possible interpretations of a matrix or is it based on a principle of mathematics?

What is x2? I did ctrl+f and only found 1 mention.
I’ll still try an answer though.
No.
Placing equations into a system does not mean the equations need to have anything to do with one another or mean anything in real life. People use matrices to solve problems because mathematicians have developed efficient representations of real world problems and methods to deal with them.
Granted, if two equations do come from the same domain, such as 2 people, each working for different hours, getting a different total pay, and assuming both of them have

It still doesn’t mean that the system of equations is solvable. To be solvable, we are assuming the pay per hour worked is the same for the 2 people representing the 2 equations. AND the data (x,y) collected for every person is accurate. This doesn’t happen in real life so people do regressions to minimise error rather than solve for exact answers.

With all that said, let’s assume all the above conditions are satisfied, and people believe that the system of equations can be solved, the result will be that we can easily read from the solution (a manipulated form of the initial matrix) what the values of the variables we are solving for are.

Yes you had to interpret it that way during matrix setup time already, not only after solving time. Maybe the term design matrix will inspire some understanding.

I wouldn’t say any equation in the system affects any other. They are all multiple observations of the same phenomena (or you believe it to be, else you won’t put them in the same system) you are solving for.

Here is a 6 part playlist on Gilbert Strang’s 2020 Vision (actually a summary) on his Linear Algebra Course: https://www.youtube.com/watch?v=YrHlHbtiSM0&list=PLUl4u3cNGP61iQEFiWLE21EJCxwmWvvek
Basically he breaks the value of his course into 2 parts, solving equations + eigen/singular value decomposing matrices, the latter of which is very important to data scientists doing feature selection and recommendation systems

Please continue to reply the thread rather than edit your 1st post. Editing your 1st post makes it harder for me and anyone else to see why we are circular referencing in our first two posts.

I was saying the same pay per hour, not total pay, but anyway please ignore my point on this. I haven’t done this mission in years, after checking it now I realised my wrong interpretation. So the question is not about two people but two jobs with different dollars/hour.

Yes.
You set up the system of equations, because you believe each equation in the system to be true, and you believe they are solvable (or maybe not, but you hope it to be, and find out after working through).
So every equation in the system, or thinking of the system as a whole, at the moment of setting up, is already True.
Then, transformations happen. At every stage of transformation, the system is still True, every equation in the system, if you break down the system in the middle of transformation, is still True. It’s just that the last state after all the transformations is the easiest to interpret stage, when you have

[1, 0, 45],
[0, 1, 2350] 

you can read the solutions for the unknowns you are solving for directly from it.

You can try thinking about the row echelon transformations, they are exactly the same as taking 1 equation and subtracting another equation from it. Information is just represented in a more concise way when you use matrices compared to a primary school student using simultaneous equation form with all the unknown variables included in every equation, but beyond representational changes, every part of the system is assumed True at every step, until proven otherwise. There are patterns/properties that allow mathematicians to conclude a certain system of equations is unsolvable. In operations research, there is the interesting study of relaxing conditions by adding slack variables to make unsolvable system of equations, solvable.

I believe I understand now. So, the answer to my question was that equations placed in a matrix are all simultaneously true.

Of course, based on the articles referenced in my original question, there could be other different but equally valid “real-world” and plain mathematical interpretations of the matrix.

The matrix operations follow mathematical rules. I read somewhere about the validity of matrix operations.The short answer is that all matrix operations represent “balanced operations”, mainly adding or dividing both sides of the equation with the same value or adding an equation to another. I feel the question has been answered. Do you have anything to add/correct?

I have nothing to add to that point about “balanced operations”.
I see matrices as just information representation tools convenient for mathematicians to notate and play with.