Here is my favorite: [Victor Prasolov, *Problems and Theorems in Linear Algebra*][1]. [Also avaliable in PDF][2]. However, I wouldn't recommend this to any undergrad without olympiad background.

Generally, writing textbooks in form of problem compendiums is distinctive for Soviet mathematics. I could name some more books of this kind (such as one on Lie algebras), but unfortunately they are all in Russian and most have never been translated.


  [1]: http://www2.math.su.se/~mleites/Prasolov/prasLinAlg/pr-linAlg-main.dvi
  [2]: http://mathsouls.110mb.com/Ebook/Math/En/Linear%20Algebra/Problems%20and%20Theorems%20in%20Linear%20Algebra.pdf