Looking for an Example of an Amusing Theorem in Linear Algebra.

[caffeinemachine]

My friend, who is a beginner in college mathematics, recently asked me to teach her linear algebra.
She has a good grip on High School math.

I am looking for an amusing theorem in linear algebra which can be appreciated by a beginner in college mathematics and at the same time arouse interest in the subject.

​

I couldn't think of anything but the Euler Rotation Theorem.

I have expressed it in the following way:

If a rigid body is moved in space from an initial orientation to a final orien-
tation, so that some particle of body is at the same point in space in both these
orientations, then infinitely many particles of the body are at the same point in
space in both the initial and the final orientations. In fact, through any ‘fixed’
particle passes a line such that all of the particles of the rigid body which lie on
this line in the initial configuration lie in the same position in space in the final
configuration.

​

I am not sure if a beginner can appreciate the theorem immediately, especially in the form I have expressed.

Can someone please suggest a better way to express this theorem in words or suggest a different theorem entirely for my purpose?
Thank you.

 

[Opalg]

That result seems to require the extra condition that the fixed point should be an internal point of the rigid body. For example, if the spherical ball $(x-1)^2 + y^2 + z^2 \leqslant 1$ is rotated about any line in the $y$-$z$ plane, then the origin will be the only fixed point of the ball.

Euler's original statement of his theorem apparently took the fixed point to be the centre of the ball.

On the more general question of amusing theorems in linear algebra, I always found it an awkward subject to teach, because it is full of dry technicalities and singularly lacking in humour. Its usefulness only becomes apparent later on, in applications to geometry, mechanics or differential equations.

 

[Deveno]

This is really an elementary example, but it can be used to exhibit how linear algebra works in a transparent way:

A farmer verifies all his livestock (chickens and cows) is safely penned up every night, by counting heads and feet. He counts 7 heads, and 18 feet, which is as it should be. How many of each animal does he keep?

Now, one can probably solve this in one's head, by trial-and-error. Nevertheless, it is a perfect example to show row-reduction/gaussian elimination, and "the inverse matrix method". It is also simple enough to show that Cramer's method works, as well, without exhorbitant calculations.

That is, in order of increasing abstraction, we have:

Systems of linear equations--->matrix algebra--->linear algebra (using linear transformations).

People often grasp the abstract when they have an appreciation of "low-dimensional" linear algebra (specifically: the vector spaces $\Bbb R^2$ and $\Bbb R^3$, which have natural physical analogues).

Furthermore, in illustrating something like the rank-nullity theorem (the importance of which cannot be understated), low-dimensional examples let one examine ALL the possibilities, leading a new student to suspect it is true "in the general case".

If your friend has had any exposure to calculus at all, you can point out that "locally" all "smooth" functions are linear ones, so that focusing on linearity is not quite so "restrictive" as it might seem, it is a reasonable approximation in the short-term. That is any (suitably nice) function is, near a given point, "almost" a matrix.

2x2 matrices are interesting, in and of themselves: it is somewhat surprising that many interesting symmetry groups, the integers, and the complex numbers can all be represented by 2x2 real matrices in ways that are not "obvious"

(for example, the representation of $\Bbb Z$ via the monomorphism:

$k \mapsto \begin{bmatrix}1&k\\0&1 \end{bmatrix}$

which has to be seen to be believed).

Linear algebra, in all its glory, requires something of a "ramp-up" to fully grasp. (Linear) mappings from the Euclidean plane to itself, is not such a big pill to swallow.

 

[caffeinemachine]

Thank you so much Deveno and Opalg for your valuable suggestions and comments.

More suggestions are welcome.

 

[ThePerfectHacker]

An "isometry" of the plane is a function $f:\mathbb{R}^2\to\mathbb{R}^2$ such that $|a-b| = |f(a)-f(b)|$ for all $a,b\in \mathbb{R}^2$. In other words, an isometry is a transformation of the plane that preserves distance. It is a well known result that all isometries are either rotations, reflections, translations, or glide-reflections. You can prove this result using linear algebra.

 

[Evgeny.Makarov]

I think the fact that the volume of a parallelepiped equals the determinant of the the tree vectors that define it is pretty amazing. I mean this weird function that combines 9 numbers in an intricate pattern ends up being equal to the volume? And then it turns out that this intricate pattern arises from simple properties: that the volume of a unit cube is 1 and, more importatly, that volume is linear with respect to each of the three vectors.