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I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen.

Every discussion section I am trying to show my students (give them as a series of exercises that we do on the blackboard) some serious math that they can understand and appreciate. For example, when we were talking about cross products, I showed them the isomorphism between $\mathbb{R}^3$ with cross product and Lie algebra $so(3)$. Of course I didn't use fancy language, but we have proved Jacobi identity for cross product, then we picked a basis in skew-symmetric matrices $3\times 3$ and checked that when you commute these three basis vectors, you get exactly the "right hand rule" for cross product. So these two things are the same.

For the next recitation the topics are 1) eigenvalues and eigenvectors; 2) real quadratic forms.

Can you, please, recommend some cool things that we can discuss with them so that they will learn about eigenvalues and forms, but without doing just boring calculations? By "cool things" I mean some problems coming from serious mathematics, but when explained properly they might be a nice exercise for the students.

Moreover, do you know if there is a book with serious math given in accessible for freshmen form (in the form of exercises would be absolutely great!)?

Thank you very much!

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You can talk about $SO_3(\mathbb{R})$ and how the eigen vector corresponding to the eigen value 1 is the axis of rotation; and how the other eigen values give the angle of rotation. – Prahlad Vaidyanathan Oct 1 '13 at 4:21
It's more on the applied side, but there are some neat examples in spectral graph theory accessible to those with only a the most basic knowledge of linear algebra. And many of the engineers in the crowd will actually need to perform data analysis in their careers. – dls Oct 1 '13 at 5:19

If you plan on talking about matrix diagonalization when you discuss eigenvalues, you could show how eigenvalues can be used to give closed forms for linear recurrences. For example, if $F_n$ denotes the $n$th Fibonacci, then $$\left( \begin{matrix} F_{n+1} \\ F_n \end{matrix} \right)=\left(\begin{matrix}1 & 1 \\1 & 0 \end{matrix}\right)\left(\begin{matrix} F_n \\ F_{n-1} \end{matrix}\right)=\left(\begin{matrix}1 & 1 \\1 & 0 \end{matrix}\right)^n \left(\begin{matrix} 1 \\ 0 \end{matrix}\right).$$ By diagonalizing, we can find a closed form for the $n$th Fibonacci, as a linear combination of the eigenvalues of the matrix $$\left(\begin{matrix}1 & 1 \\1 & 0 \end{matrix}\right).$$

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Thank you! I think this is a really cool example! Unfortunately, the textbook we are using (Hubbard--Hubbard) already has this example (this is the only one non-trivial example in the section). I was actually quite surprised to see this example as motivational for studying eigenvalues. – Sasha Patotski Oct 1 '13 at 4:47
A professor gave us this exercise an entire semester before we even knew the word "eigenvalue" and it was fascinating to me. – Jack M Oct 1 '13 at 11:47

I myself did not study graph theory yet but I do know that if you consider the adjacency matrix of a graph, then there are interesting things with its eigenvalues and eigenvectors.

You can also show how to solve a system of ordinary differential equations of the form $$ {x_1}'(t) = a_{11}x_1(t) + a_{12}x_2(t) + a_{13}x_3(t)\\ {x_2}'(t) = a_{21}x_1(t) + a_{22}x_2(t) + a_{23}x_3(t)\\ {x_3}'(t) = a_{31}x_1(t) + a_{32}x_2(t) + a_{33}x_3(t) $$ using linear algebra. Let $$ x(t) = \begin{pmatrix} {x_1}(t)\\ {x_2}(t)\\ {x_3}(t) \end{pmatrix}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}. $$ Then the system can be written as $x'(t) = Ax(t)$. You can diagonalize (assume its possible) to get a invertible matrix $Q$ and diagonal matrix $D$ so that $A = QDQ^{-1}$. Then $x'(t) = QDQ^{-1}x(t)$ or $Q^{-1}x'(t) = DQ^{-1}x(t)$. Let $y(t) = Q^{-1}x(t)$. Then $y'(t) = Dy(t)$. Since $D$ is diagonal, this is now a system of equation which is independent of each other which can be solved easily. I thought this was a nice application. This was in my linear algebra book by Friedberg, Insel and Spence.

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Personally, I remember my linear algebra experience well because it taught me about rings and modules. Unfortunately, this is a dangerous strategy because not so many people are attracted by this abstraction :)

However, I do remember enjoying some of the projects in Numerical Linear Algebra by Trefethen and Bau being very fun, direct applications of linear algebra. Specifically, there was a project about storing images, which used successively improving low-rank approximations. It was interesting to see the image evolve. Solving least-squares problems might also be interesting. The projects do a good job of making eigenvalues and other abstract concepts seem very practical.

To display this stuff, you'd probably need access to MATLAB. Actually, half the fun for the student is doing the project with their own hands. Having MATLAB do all the computations relieves the student form having to do so: they can just savor the results! You might be able to sketch out a few theoretical excursions to explain the results, using the projects just to motivate the subject matter.

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I think it's interesting that the eigen-decomposition of the adjacency matrix of a (combinatorial) graph leads to "harmonious" geometric realizations of the graph.

See my answer here.

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Maybe these won't be "serious" enough for you, but I think the most interesting uses of eigensystems are:

(1) Understanding/classifying quadric surfaces.

(2) Moments of inertia (principal axes)

(3) Principal component analysis

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While the result will (presumably!) be beyond your students' capabilities to prove, if you're covering quadratic forms you might introduce them to the 15 and 290 Theorems (both for the out-of-left-field result itself and also to distinguish between 'integral' and 'integral-valued' forms). You could also introduce them to the topograph as a tool for investigating equivalences (and various properties - minimal values, positivity, etc.) of quadratic forms - I highly recommend The Sensual (Quadratic) Form as extra reading if you have the opportunity for such.

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This is a really cool stuff! I don't think I will be able to say anything about the proofs, but I definitely will tell them about these surprising results! – Sasha Patotski Oct 1 '13 at 18:24

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