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I was thinking today applications of $\textbf{Ax}=\textbf{b}$ where $\textbf{A}\in\mathbb{R}^{m \times n}$. Specifically, I am interested to know what applications one might give to students, who don't know anything about the topic, when the dimension is large. For instance, I would typically introduce all the necessary apparatus, e.g. multiplication or linear combination, row-reduction, to solve the problem and present the geometric implications for $\textbf{A} \in \mathbb{R}^{3 \times 3}$. However, if you wanted to introduce numerical techniques, motivated by large coefficient data, then what tangible and accessible problem might you lay out for a young student? Something analogous to the intersection of planes in space, which even a young student could meaningfully fiddle with outside of an instructional setting.

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    $\begingroup$ Young. Say pre-calculus. The examples you mention are the ones that come to my mind as well, but are inaccessible without a calculus sequence and, often, more coursework. $\endgroup$ – user153764 Mar 14 '16 at 17:43
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    $\begingroup$ Of course, some of us have 40+, or 60+, year-old students who are yet working to reach the level of precalculus. $\endgroup$ – Daniel R. Collins Mar 15 '16 at 11:51
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One possible class of examples of high dimensional systems is discretizations of infinite dimensional ones. Differential equations, as suggested in the comments, are one option, but let me propose another one that hopefully seems interesting and useful to students. This is a linear problem that is solved several times every day in hospitals around the world.

In X-ray tomography one attempts to construct a three dimensional image of an object from measuring the attenuation of X-rays passing through it. The problem can — under ideal circumstances — be easily reduced to recovering a function in $\mathbb R^n$ (with $n=2$ or $n=3$) from its integrals over lines. The function is physically the (non-constant) attenuation coefficient. The key object here is the X-ray transform, an integral transform that takes a function on the Euclidean space to a function over the set of all lines in that space. The value of the X-ray transform of a function at a line is the integral of that function over that line.

The question is then whether the X-ray transform is injective. If it is, ideal measurements indeed uniquely determine what we wanted to know. The X-ray transform is linear.

It takes a while to set up a theory of the X-ray transform with proper function spaces and such. But that is not necessary. You can start building a model starting from the simpler end. For example, take a $3\times3$ grid with unknown numbers in each square (compare to a sudoku) and take some lines through that square. Those numbers are the values of the unknown functions in the squares, and outside the bigger square the function vanishes. Given your set of lines (pick horizontal, vertical and diagonal ones), can you determine the nine numbers from the integrals? You need to have at least nine lines, but more can make the problem easier.

If we want to see inside a human patient, nine pixels is not quite enough. What resolution should we have? How many lines would we need? Will the X-ray transforms still be injective?

The students can't tackle the problem by hand in realistic scales, but they can play with it with small resolution and learn to appreciate the problem. If the course (or your students) is numerically oriented, you can have them play with a real X-ray tomography dataset. The walnut looks a bit like the human brain inside a skull, doesn't it?

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Balancing Chemical Equations

Balancing chemical equations is an effective and useful application, especially for students who are also taking chemistry. This article from the Journal of Natural Sciences Research gives a detailed explanation and provides multiple examples. From the conclusion of that article:

This allows average, and even low achieving students, a real chance at success. It can remove what is often a source of frustration and failure that turns students away from chemistry. Also, it allows the high achieving to become very fast and very accurate even with relatively difficult equations.

The practical superiority of the matrix procedure as the most general tool for balancing chemical equations is demonstrable. In other words, the mathematical method given here is applicable for all possible cases in balancing chemical equations.

It seems like a cumbersome method, but students get the knack of writing the matrix directly from the chemical equation, and using a calculator to put the augmented matrix into rref allows for students to balance any equation in about a minute, an incredible improvement over the guess-and-check method that most chemistry teachers employ. It's an invaluable method that I teach all my students that are also in AP Chemistry, since they are allowed to use their calculator on the exam.

This site lists about 260 unbalanced chemical reactions from which you can pick exercises.

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A relatively low dimensional example is the problem of oil mixing: when preparing a certain quality of gas to be sold, an oil company mixes several oils of various composition which have been extracted and refined, and aims to get a certain composition. The question is in which ratio to mix the oils to get the wanted gas (I hope I got the vocabulary right, I am not a native English speaker).

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