Problem book on linear algebra

My first suggestion would have been Schaum's outline. However since you have gone through that already, another book I am quite fond of (which I think covers a good portion of the topics you mentioned) is "Linear Algebra Problem Book" by Paul Halmos:

http://www.amazon.co.uk/Algebra-Problem-Dolciani-Mathematical-Expositions/dp/0883853221

It possesses a whole range of solutions on a range of problems in linear algebra. The chapters cover topics from scalars and vectors, to canonical forms and inner product spaces. I'd recommend it.

You may find the following "Problems and Theorems on Linear Algebra" by V. Prasolov:

http://www2.math.su.se/~mleites/books/prasolov-1994-problems.pdf

How do I prove this combinatorial identity using inclusion and exclusion principle?

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