Applications of systems of linear equations

Question

Sorry if this questions is overly simplistic. It's just something I haven't been able to figure out.

I've been reading through quite a few linear algebra books and have gone through the various methods of solving linear systems of equations, in particular, $n$ systems in $n$ unknowns. While I understand the techniques used to solve these for the most part, I don't understand how these situations present themselves. I was wondering if anyone could provide a simple real-world example or two from data analysis, finance, economics, etc. in which the problem they were working on led to a system of $n$ equations in $n$ unknowns. I don't need the solution worked out. I just need to know the problem that resulted in the system.

Answer 1

One of the most frequent occasions where linear systems of $n$ equations in $n$ unknowns arise is in least-squares optimization problems. Let us look at an example. Let's say that we are studying two physical quantities $y$ and $x$ and we conjecture that $y$ is a second order polynomial function of $x$, i.e. $y=\alpha x^2 + \beta x + \gamma$ for some real numbers $\alpha$, $\beta$, $\gamma$ that are unknown. Let's say now that we perform experiments and obtain measurements $(x_1,y_1) \cdots (x_{100},y_{100})$. Applying the polynomial model on the measurements yields $y_i=\alpha x_i^2 + \beta x_i + \gamma$ for $i=1, \cdots 100$ or in matrix form $X k=y$ where $k=[\alpha \, \, \beta \, \, \gamma]^T$, $y=[y_1 \cdots y_{100}]^T$ and the $i^{th}$ row of $X$ is the row vector $[x_i^2 \, \, x_i \, \, 1]$. Now, as you might observe, we have $100$ equations in $3$ unknowns, i.e. our linear system $X k=y$ is overdetermined. Practically speaking, this system is consistent (i.e. it has a solution) only if indeed $y$ is related to $x$ via a second order polynomial equation (i.e. our conjecture is true) and additionally there is no noise in our measurements. So assume that none of the above two conditions is true. Hence the system $X k=y$ will not in general have a solution and one might consider finding a vector $k$ that instead minimizes $||X k - y||_2^2$, i.e. the square of the error. Then the solution of this optimization problem is the solution to the $3 \times 3$ system $X^T X k = X^T y$. This formulation comes up all the time in engineering, e.g. in signal prediction. So, least squares problems lead to square (i.e. $n \times n$) linear systems of equations.

Answer 2

I have no knowledge of finance or economics, but problems in physics and engineering often give rise to linear systems.

For example, say you have a complicated network of resistors, and you apply a potential difference to two junctions in the network (connect them to the ends of a battery, say). What is the current going through each resistor? If the network can be split up into series and parallel combinations of resistors, it's easy to solve, but in general, it's not possible to do this. Then the only way to find the solution is linear algebra: you have $n$ unknowns, the relative potentials at all the other junctions, and $n$ equations, from Kirchhoff's current law at those junctions, and you can solve this system to find the unique solution.

If you don't think this is a real-world problem, you should listen to MC Frontalot: people call him up in the middle of the night to ask him the impedance of a resistor icosahedron.

Answer 3

A package of gummi bears contains 20 grams of sugars, 5 grams of fat, and 1 gram of proteins.

A package of butter contains 6 grams of sugars, 15 grams of fat, and 2 gram of proteins.

A chicken contains 2 grams of sugars, 4 grams of fat, and 12 gram of proteins.

(The numbers are fake.)

Joe has bought some gummi bears, butter, and chickens. In total, he ate it and it contained 1 ton of sugars, 1 ton of fats, and 1 ton of proteins. How many packages of gummi bears, butter, and chickens did he buy?

Answer 4

Here is an example from cost accounting:

In Chapter 15 of Cost Accounting: A Managerial Emphasis, 14^th ed., by Horngren, Datar, and Rajan, one of the sections covers a method of allocating costs of support departments such as Personnel and Legal to operating departments such as the refrigerator and dish washer manufacturing divisions of a kitchen appliances manufacturer. Special consideration needs to be given for the common scenario where support departments support each other. For example, the Personnel department may support the Legal department by hiring paralegals and attorneys while the Legal department might support the Personnel department by evaluating compensation plans and interpretting regulatory requirements.

The most justifiable method presented in the chapter is called the "reciprocal method" or "matrix method". It uses a system of linear equations to calculate the total cost of each support department, factoring in reciprocal support, so that the total cost can be allocated to the operating departments.

For example, assume the following facts:

• The Personnel Department's budgeted fixed costs are $1,200,000, not including reciprocal support costs.
• The Legal Department's budgeted fixed costs are $2,000,000, not including reciprocal support costs.
• Personnel Department costs are allocated on the basis of recruiter hours.
• Legal Department costs are allocated on the basis of attorney hours.
• The Personnel Department is budgeted to utilize 5% of the Legal Department's budget for attorney hours.
• The Legal Department is budgeted to utilize 10% of the Personnel Department's budget for recruiter hours.

Let $P_F$ be the Personnel Department's total fixed costs and $L_F$ be the Legal Department's total fixed costs. Then:

$$ \begin{align} P_F &= \$1,200,000 + 5\% \times L_F\\ L_F &= \$2,000,000 + 10\% \times P_F \end{align} $$ We have: $$ \begin{align} L_F &= \$2,000,000 + 10\% \times (\$1,200,000 + 5\% \times L_F) \\ 99.5\% \times L_F &= \$2,120,000 \\ L_F &= \$2,130,653.27 \end{align} $$ $$ \begin{align} P_F &= \$1,200,000 + 5\% \times \$2,130,653.27\\ &= \$1,306,532.66 \end{align} $$ The related matrix is: $$ \begin{bmatrix} 1 & -0.05 \\ -0.1 & 1 \end{bmatrix} $$

The system of linear equations is larger with more support departments.

Answer 5

Here are two neat exmaples I found while researching the topic for a class I'm teaching:

http://www.ohiouniversityfaculty.com/youngt/IntNumMeth/lecture9.pdf (The example deals with systems of equations in statics problems in mechanical engineering; presumably, an introductory text on statics for mechanical engineers might contain more real-life examples by the truckload?)

https://www.researchgate.net/publication/339843126_Balancing_Chemical_Equations_Using_Gauss-Jordan_Elimination_Aided_by_Matrix_Calculator (Just about every introductory [and/or advanced?] chemistry textbook should contain real-life examples by the truckload.)

Hope this helps.