Tricky linear algebra/abstract algebra problems sought

This is the linear algebra/abstract algebra equivalent of this question I recently asked. I'm working through some linear and abstract algebra textbooks and of course, being texts primarily aimed at introducing a concept, the problems in them are far from difficult once you've memorized/intuitively understood/become familiar with the corresponding theorems/concepts. These problems are great for getting familiar with material, but I can't help but feel that I'm not developing problems solving skills.

Allow me to give analogy. Suppose some students were taught elementary geometry and angle chasing (cyclic quadrilaterals, centers of triangles, etc.). Now suppose they were asked to solve IMO 2006 Problem 1. There might not be much success. And this problem doesn't require advanced barycentric coordinate techniques or inversion or Miquel points or anything else. Just angle chasing.

Thus I seek linear/abstract algebra problems that are to the elementary concepts in linear/abstract algebra as IMO 2006 Problem 1 is to the elementary concepts in Euclidean geometry. Or perhaps more generally as Olympiad-type problems are to the concepts required to solve them. They're elementary (i.e. you don't need big theorems/concepts), but non-trivial nonetheless.

Edit: I would also like to add that I am not looking for problems where linear/abstract algebraic concepts have been used in some other field. For example, we have the following classic: Let $$\{A_{i}\}_{i be a finite family of $$m$$ finite subsets of $$[n]=\{1,2,\dots,n\}$$, each with odd cardinality, such that for all distinct $$i,j$$, $$|A_{i}\cap A_{j}|$$ is even. This is solved by representing each $$A_{i}$$ by a "characteristic" vector $$v_{i}\in\mathbb{F}_{2}^{n}$$, where the $$j$$th coordinate is $$1$$ iff $$j\in A_{i}$$, and showing that these vectors are independent.

I love this problem. I think it's brilliant. But it's not what I'm looking for, since I'm not trying to use algebra to further my abilities in other areas, I'm instead trying to further my abilities in algebra.

• If the problem in linear algebra is hard then it's probably not a problem of linear algebra. For example, pick any random $n \times n$ matrix and ask to calculate its eigenvalues. Very hard problem. However, the difficulty is simply factoring. Linear algebra is easy. Feb 10 '20 at 2:09
• @JamesS.Cook Ok, I suppose you would know better than I. Is it recommended, then, if I know the theorems and definitions and am comfortable with doing linear algebra presented in the standard texts that I move on to more abstract algebra/other areas? I don't want to go into a field able to understand what I'm studying but unable to come up with any proofs or ideas because I lack a strong foundation. Perhaps areas which use linear algebra like algebraic topology do not really have this problem? Feb 10 '20 at 2:30
• since you mentioned IMO quite a bit, you may want to comb through past Putnam problems, there are some that explicitly fit and some that implicitly fit kskedlaya.org/putnam-archive Feb 10 '20 at 6:52
• Well there are so many excellent texts in Linear Algebra, you can find nearly endless advice. Maybe "The Linear Algebra a Beginning Graduate Student Ought to Know" for usual proof and computation advice is a good resource (I have it recommended by a well-educated friend) and I know Golub and Van Loan's "Matrix Computation" gets you a deep slice of that which lies beyond the basics. Know spanning, LI, finding a basis and characterizations as null spaces deeply. Theoretically, the isomorphism theorems are what are often used in theoretical applications. Often, the trick is constructing a basis. Feb 11 '20 at 2:03