Make sure you pick up Axler's "Linear Algebra Done Right". It is my definite pick for someone entering the topic. And the second edition has a solutions manual

Make sure you pick up Axler's "Linear Algebra Done Right". It is my definite pick for someone entering the topic. And the second edition has a solutions manual floating around, which is very useful for practice (more importantly, every theorem proof in a math book is the solution to a practice problem...).