Sadly, this is not actually the way linear transformations work, from xkcd.
This semester I’m teaching Linear Algebra for the first time, so naturally, I am constantly on the prowl for all of the linear algebra resources the internet has to offer. To begin with, I’m using a free online textbook called Linear Algebra Done Wrong by Sergei Treil. I’ve found that it’s a bit…intense. As a person who understands linear algebra the book is very nicely written and has a logical presentation and abundant clever examples. But for a person who has never seen linear algebra, well, let’s just say it’s a bit like diving into the deep end with no floaties on while someone shoots you with a paintball gun.
Consequently, this semester has left me foraging the world wide web for supplementary resources to help my poor flailing floatieless students as they try to navigate the waters of vectors and matrices.
A great place for students to begin if they are totally lost is a series of wonderful YouTube videos called the Essence of Linear Algebra, from 3Blue1Brown. The animations really help to bring out some of the geometric intuition behind vector spaces, which can seem abstract (and sometimes totally impenetrable!) to students seeing them for the first time. Of course Khan Academy also hosts a linear algebra series, but my students haven’t found them as helpful.
Don’t like the dog? You can also toggle between a cat and a mouse. Screenshot from Wolfram Demonstrations Project.
On the theme of helpful animations, which teaching linear transformations, I found some really great demonstrations on Wolfram that let you transform a dog, more specifically, a Scottish Terrier, by a personalized 2×2 matrix. You can stretch, flip, and shear the Scottish Terrier by changing the values in the accompanying matrix. Somehow this is way more convincing than just drawing pictures and waving your hands around. The Wolfram Demonstrations Project is packed with great demos for transforming vector spaces, and you can share your own.
Finally, when I ask former math majors what most mystified them about Linear Algebra I almost always hear something about eigenvectors. It’s shocking how many students get in and out of Linear Algebra and have no intuitive idea what an eigenvector or eigenvalue are. And I’m not passing judgement here. When I took Linear Algebra as an undergraduate I was in the same boat! I knew how to compute them, but I had no idea what I was really looking at.
Not just a beautiful pirouette, but also a great example of a linear transformation with eigenvector e2 and eigenvalue 1.
Luckily, we have Steven Strogatz to the rescue with a most concise and intuitive explanation of eigenvectors and eigenvalues. He compares a linear transformation of 3-dimensional space to snapshot of a dancer, arms outstretched spinning in a pirouette-like motion. Her arms (a vector in the x direction) are moving, her gaze (a vector in the z direction) is moving, but the leg she’s spinning on (a vector in the y direction) stays fixed. This fixed vector is an eigenvector. And if she comes down of her pointed toes, then there is some element of scaling. This is an eigenvalue.
Mathew Simonson, who wrote about his own “eigenightmares” for the AMS Grad Student blog, proposes a spiral approach to pedagogy to combat those eigenfears. Students learn early on to express linear transformations as matrices, at this point they already can get some sense of eigen-type behavior just by acting on a simply figure in a vector space. Say, maybe, a nice little Scottish Terrier. In this way, students can see that eigenvectors are happening, before formally knowing what they are. This puts the intuitive before the formal, which I like.
Do you have any favorite online resources for teaching or understanding Linear Algebra? Let me know on Twitter @extremefriday.
“He compares a linear transformation of 3-dimensional space to a dancer doing a pirouette. Her arms (a vector in the x direction) are moving, her gaze (a vector in the z direction) is moving…”
This is (a) incorrect and (b) not what Strogatz said.
For (a), in a pirouette the dancer’s arms are curved (as in your animated gif). They don’t represent vectors. (And in a non-partnered pirouette, they don’t act as in a rigid motion.) Also, a dancer’s gaze is fixed before/after a pirouette so in terms of a transformation it would be fixes. Indeed, if we want to think of a pirouette as a linear transformation (which it’s not naturally) then we would consider the start and end of the rigid portion of a pirouette, which are identical so the pirouette would be an identity transformation. One could also think of a partial pirouette, but that’s not what was described at all.
For (b), Strogatz basically said to consider a dancer standing in first position with arms in second position, and then rotate the person by 90 degrees; now the arms *do* represent vectors, and they move but the axis through the core of the body and floor does not move.
Sarah-Marie, thanks for the comment. I guess it’s my lack of understanding of what a pirouette is that obscured my explanation. I updated the post 🙂 Thanks of reading!
I have put some examples of area-preserving and angle-preserving maps in the Abstractmath.org article “Images and metaphors for functions” at http://www.abstractmath.org/MM/MMImagesMetaphorsFunctions.htm#mappreservationproperties. If you have access to Mathematica, there are more in the notebook Functions and shapes.nb at http://www.abstractmath.org/MM/Mathematica/Function Representations/Functions and shapes.nb. Both of these are licensed under Creative Commons.
I like to think of eigenvectors as generators of one-dimensional invariant subspaces. In some sense, to me there are two key ideas involved: 1) eigenvectors only make sense for endomorphisms, not arbitrary linear maps and 2) they help us choose the “best” basis to describe a given endomorphism.
My favorite examples are using diagonalization to compute very high powers of matrices, and explaining the relationship between Fibonacci numbers and golden ratio.
I find the “example” with the ballerina confusing rather than enlightening, but then I find all physics extremely confusing.