Cool mathematics I can show to calculus students. 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!

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.
