I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in discussion section.
In our class, we cover the basics (linear transformations; matrices; subspaces of $\Bbb R^n$; rank-nullity), orthogonal matrices and the dot product (incl. least squares!), diagonalization, quadratic forms, and singular-value decomposition.
Showing my ignorance, the only application of these I know is the one that was presented in the linear algebra class I took: representing dynamical systems as Markov processes, and diagonalizing the matrix involved to get a nice formula for the $n$th state of the system. But surely there are more than these.
What are some applications of the linear algebra covered in a first course that can motivate the subject for students?
Best Answer
I was a teaching assistant in Linear Algebra previous semester and I collected a few applications to present to my students. This is one of them:
Google's PageRank algorithm
This algorithm is the "heart" of the search engine and sorts documents of the world-wide-web by their "importance" in decreasing order. For the sake of simplicity, let us look at a system only containing of four different websites. We draw an arrow from $i$ to $j$ if there is a link from $i$ to $j$.
The goal is to compute a vector $\underline{x} \in \mathbb{R}^4$, where each entry $x_i$ represents the website's importance. A bigger value means the website is more important. There are three criteria contributing to the $x_i$:
Each website has exactly one "vote". This vote is distributed uniformly to each of the website's outlinks. This is known as Web-Democracy. It leads to a system of linear equations for $\underline{x}$. In our case, for
$$P = \begin{pmatrix} 0&0&1&1/2\\ 1/3&0&0&0\\ 1/3& 1/2&0&1/2\\ 1/3&1/2&0&0 \end{pmatrix}$$
the system of linear equations reads $\underline{x} = P \underline{x}$. The matrix $P$ is a stochastical matrix, hence $1$ is an eigenvalue of $P$. One of the corresponding eigenvectors is
$$\underline{x} = \begin{pmatrix} 12\\4\\9\\6 \end{pmatrix},$$
hence $x_1 > x_3 > x_4 > x_2$. Let
$$G = \alpha P + (1-\alpha)S,$$
where $S$ is a matrix corresponding to purely randomised browsing without links, i.e. all entries are $\frac{1}{N}$ if there are $N$ websites. The matrix $G$ is called the Google-matrix. The inventors of the PageRank algorithm, Sergey Brin and Larry Page, chose $\alpha = 0.85$. Note that $G$ is still a stochastical matrix. An eigenvector for the eigenvalue $1$ of $\underline{x} = G \underline{x}$ in our example would be (rounded)
$$\underline{x} = \begin{pmatrix} 18\\7\\14\\10 \end{pmatrix},$$
leading to the same ranking.