this one is a bit inverted!
So I am busy doing an advanced undergrad course in Linear algebra, and it is going very well, the problems in the book seem fairly routine. To be able to see if I am any good at this "math thing", I was wondering if some well-versed mathematician could pose a fairly "deep" question in linear algebra that can be proven without anything too advanced from other courses, but is still challenging and requires some deep stuff/maturity. This way I can try prove it by myself. If I can't, I should probably abandon math early which can be a good thing too...
example of material covered: finite-dimensional vector spaces, inner products, linear transformations, dual spaces and pullbacks, operator algebras, operator representations/matrices, determinants and spectral decomposition.