Text recommendation for introduction to linear algebra

I have learned linear algebra before, but with my right brain - basically nothing left, so, anyone can recommend me good textbook so I can learn it again?

The text should subject to the conditions listed below.

Currently, I have found, from other questions asked here, Linear Algebra Done Right by Sheldon Axler, and Advanced Linear Algebra by Steven Roman.

It will be used for preparing an exam, (Don't take me wrong, I like math), and the later one has an "advanced" in its name, so I'd better pass it.

The first one is good, but got a little greedy, want a better one~

Video course is OK too.

Conditions:

• It should be a introduction, but not too elementary.
• Don't begin with linear equations, matrices or determinant. (The one I used did so, these stuffs are so boring, and the experience is really terrible.)

• It should cover: (I don't really believe this condition can be satisfied, so, the more the better)

  • Matrices (inverse, rank, Jordan canonical form etc.)
  • Determinant
  • Kernel space, dimension of vector space and quotient space
  • Linear map and its relation with matix
  • Other "common topics" a LA introduction text should cover, and have some "famous results". (Don't let me define "common topics", I assume the answerer knows so much about LA that he knows which topics should be covered, and which should not be. Same for the "famous results" :-)
• Don't care real world applications and numeric computation.
• Do it in a more "mathematical"/modern/abstract way, i.e., every term is formally defined, proofs are rigorous, etc..
• Prefer formal proofs than intuitive ones.
• Prefer "proof-like" proofs than those in a computation style.
• Having good exercises is a bonus.
• Doesn't have, or has little contents about group, ring, etc.. (I have limited time)
• Clear written.

Thanks.

I suggest Schaum's Outline of Linear Algebra by Seymour Lipschutz. It is full of solved exercises which you can find out what is going on in Linear algebra intuitively.