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:
- Two jobs are represented as equations where
y = total pay,
x = hours workedand the constant
c = base pay.
- 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.
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.
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?
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”?
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?