# Fun linear algebra problems

While TAing a linear algebra class last semester, I discovered that there aren’t many resources for good problems in linear algebra. I only found a few problems over the semester that I was really happy going over in section (in that they were both the right difficulty, and exciting). I’ve collected them here in case anyone else finds them cool or useful.

(Incidentally, this seems like potentially a neat project—a collection of problems in math or other subjects rated for difficulty, prerequisites, fun, etc. with hints and solutions. Does this exist?)

Update: now with solutions to 1 and 2!

1. Let $\mathbb{F}$ be the finite field of $p^n$ elements. Let $V$ be a $k$-dimensional vector space over $\mathbb{F}$.

1. How many linear transformations $T: V \to V$ are there?
2. How many of these are invertible?
3. How many are of determinant 1?
2. Let $f_n$ be the $n$th Fibonacci number.

1. Find a linear transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$ such that $T(f_{n-1}, f_{n}) = (f_n, f_{n+1})$ for all $n$. Write down the matrix of $T$ with respect to the standard basis.
2. Find an algorithm for determining the $n$th Fibonacci number in $O(\log n)$ time rather than the typical $O(n)$ (assuming addition and multiplication are $O(1)$).
3. What are the eigenvalues of $T$?
4. Derive the closed-form expression for the $n$th Fibonacci number.
3. In an $n$-dimensional real inner product space, find the size of the largest possible set of vectors $\{v_1, \dots, v_n\}$ such that $\langle v_i, v_j\rangle$ is strictly less than 0 if $i \ne j$.

4. Let $A$ be a matrix. We say the order of $A$ is $k$ if $k$ is the smallest integer such that $A^k = I$.

1. Show that is $A$ is a $2 \times 2$ integer matrix of finite order, then $A^{12} = I$.
2. Let $M$ be the set of $n \times n$ integer matrices of finite order. What is the smallest $k$ such that if $A \in M$, then $A^k = I$?

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DirePangolin

This is a fun problem I tackled last year: classify all square matrices $A$ and $B$ such that $A^{-1}+B^{-1}=(A+B)^{-1}$. I can’t promise its appropriate for intro linear algebra students though!

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Oms

These problems look good but I barely know any linear algebra. So I’d love to see some solutions. (I’d learn at a much faster rate reading through solutions than slowly working through these using Wikipedia and Google.)

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Oms

These problems look good but I barely know any linear algebra. So I’d love to see some solutions. (I’d learn at a much faster rate reading through solutions than slowly working through these using Wikipedia and Google.)

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Ben

Oms, I was thinking about posting some solutions! The only thing is they’re much more annoying to write up than the problems :P Working on ‘em though!

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Ben

Solutions to 1 and 2 are up! (Link in the post.) Sorry it took me so long. Are they reasonably intellegible?

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Ari GP

Hey Ben, if you’re looking for a collection of worthwhile problems, look no farther than physics professor David Morin. He is obsessed with good problems and has, at the very least least, tremendous repositories of solved problems in Mechanics, E&M, and Wave Topics. I assume he has boxes and boxes of good math problems as well. He’s a great guy too and I’m sure he’d love to chat.

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Ben

Thanks, Ari! I got to appreciate some of David Morin’s mechanics problems during Physics 16, but I didn’t realize he collected E&M and wave problems as well! I’ll definitely keep hiim in mind.

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Tarn

(Incidentally, this seems like potentially a neat project – a collection of problems in math or other subjects rated for difficulty, prerequisites, fun, etc. with hints and solutions. Does this exist?

Please, good god let this exist! Especially for calc and pre-calc; I was subjected to truly awful problems in pre/calc

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