Working through some introductory problems on matrix powers

I saw a really neat problem in Strang’s Linear Algebra book earlier this week:


Tonight I had my son work through them on camera. These problems bring together ideas not just from linear algebra, but also from a high school algebra class.

Here’s his work on the first problem:

Here’s his work on the 2nd problem – this one is a fun surprise. The numbers don’t get big at all. In fact, this matrix has powers that are the identity matrix:

Here’s the third problem – a lot of the work in this problem is him remembering how to multiply complex numbers. I really like this problem because it brings in quite a bit of math from outside of linear algebra:

Here’s the last problem which is another fun surprise. We change one entry of the matrix in the previous problem by a tiny amount, and the powers of the matrix behave in a completely different way:

Sharing the Eigenvectors from Eigenvalues paper with my son

Yesterday I saw a neat tweet from Natalie Wolchover:

I was excited about the result when I first read Wolcover’s original article, but even more excited about the new paper as, by incredibly lucky coincidence, I’m covering eigenvalues and eigenvectors with my older son right now!

The paper gives a simple example of the “eigenvectors from eigenvalues” formula using this matrix:


Yesterday I had my son compute the eigenvalues and eigenvectors for this matrix, which is a nice exercise for someone who learned about those ideas two days ago! Today we tried to use the formula from the paper.

We began by looking at the formula and discussing the 3×3 matrix:

Next I had him work through the standard calculation for one of the eigenvectors:

Before moving on to the final formula, we needed to get some eigenvalues for one of the special submatrices in the formula. Unfortunately we had a little calculation goof that took a minute to find, but we eventually got the right answers:

Finally, we worked through one example of calculating the value for a component of one of the eigenvectors. This part probably could have been done a bit better by us, but live math isn’t always perfect!

I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students. It was really fun to share these ideas with my son tonight!

Sharing basic ideas about Cramer’s Rule and Fourier Analysis with my older son

My older son is studying linear algebra out of Gil Strang’s book this year. Currently he’s in the chapter on determinants and we’ve spent the last couple of days talking about Cramer’s Rule.

As we talked about the proof of Cramer’s Rule, I was struck by how similar the ideas were to the ideas used in Fourier analysis. This morning we had a fun discussion showing how the ideas are connected.

I started by asking him to talk about Cramer’s Rule. He did a nice job, especially since his knowledge about this rule is only a few days old:

Next we played around on Mathematica with a 4×4 example and found that the solutions you get from Cramer’s Rule do indeed match the solutions you get from other methods!

Next I gave a really short introduction to a problem that initially seems very different, but has a lot of the same mathematical ideas hiding in the background -> pulling a signal out of noise:

Finally, we went back to Mathematica to play with a few examples of signals hiding in noise. We saw how the ideas from Fourier Analysis could often pull out the signal even though it wasn’t obvious at all that a signal was hiding in our data in the first place!

A linear algebra follow up to our probability project

My older son is learning linear algebra right in Strang’s Linear Algebra book. Over the weekend we did a fun project on problem number #1 in Frederick Mosteller’s probability challenge book and I wanted to show him a neat follow up involving linear algebra today.

The original project is here:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today what I wanted to show my son is how you solve the recurrence relation for the terms we found in the original project. It is a fun linear algebra example.

We started by reminding ourselves what the terms were and then looking at the recurrence relation we found on the Internet Sequence Database:

By the end of the last video we had a cubic equation written down and in this video my son worked to find the roots of that cubic:

Now we looked at linear combinations of the powers of the roots that we found in the last video to see if we could find the general solution. The general solution involved a matrix equation:

We did not solve the matrix equation by hand, but rather went to Mathematica to save time.

Once we solved the equation we saw that we could find any term in the sequence that we wanted!

Intro to Linear Algebra

Having finished a single variable calculus class with my son this school year, I’ve been thinking about what to do next. Probably the next step is going to be linear algebra and we’ve been watching a few of Grant Sanderson’s “Essence of Linear Algebra” videos to get a feel for the subject.

Today I wanted to have a short and introductory talk about vectors with my son, and I had two goals in mind. The first was to show some ideas about (for lack of a better phrase) thinking in vectors rather than thinking in coordinates. The seconds was just sort of a fun introduction to the dot product.

So, I started with a simple introduction to vectors that he’s seen a bit of via the Grant Sanderson series:

Finding a vector representation for the 2nd diagonal of the parallelogram we’d drawn was giving him some trouble, so we took a deeper dive here. I’ve always thought that the equation for the 2nd diagonal was non-intuitive, so I gave him plenty of time to make mistakes and work through the ideas until he found the answer:

Finally, I did a simple introduction to the dot product and we calculated the angle – or the cosine of the angle – between a couple of vectors as a way to show how some ideas from linear algebra help solve seemingly complicated problems:

So, next week I’m having him watch a few more of Grant’s videos while I’m away on a work trip. We’ll get going on linear algebra the week after that.