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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.

BR Frederick

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Well, if you want to challange yourself, and learn Algebra at the same time, then you have plenty of options. You could google 'putman algebra' or 'putnam linear algebra.' These don't prove 'deep' results, but are considerably harder than most of the problems in most of the text books. You could also teach yourself through Artin or Dummit and Foote. Both books have great exercises and cover LOTS of essencial material for anyone interested in Math. – Chris Dugale Sep 23 '13 at 22:42
@ChrisDugale Thanks, I was planning about buying Dummit soon in fact. It is just that I am stressed about my choices, it feels like I should establish early on whether I am capable of deeper proofs. The problem I have with Putnam problems is that they are not very general, and probably constructed for specific situations. Having said that, I might give them a look-see :). Thanks for your reply, and yes Dummit contains 'everything' ! – Frederick G Sep 23 '13 at 22:48
Here is a book you can find useful: Prasolov, Problems and Theorems in Linear Algebra – Artem Sep 23 '13 at 22:55
@Artem Thanks, will check it out! – Frederick G Sep 23 '13 at 22:57
Linear algebra was what launched me into ring theory, so you might consider an abstract algebra book. I never used dummit and foote ever. I mainly consulted hungerford, isaacs and grillet. – rschwieb Sep 23 '13 at 23:47

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