Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic) I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern differential and riemannian geometry(for example, Jeffrey Lee "Manifolds and differential geometry").
I would like to chose a book on theoretical linear algebra to cover prerequisites for these. 
That is, I don't need a linear algebra for applications in science. I don't need it for its own sake or history of mathematics. I want to study modern geometry and analysis - real and complex analysis, analysis on manifolds, differential geometry, riemannian geometry, complex algebraic and analytic geometry.
I came across a few books on my search, but I'm not sure if they are the best option for my goals.
Axler "Linear Algebra Done Right" - this book seems to have a different goal in mind. It avoid determinants as well as works only over $\mathbb{R}$ and $\mathbb{C}$. Do you need linear algebra over an arbitrary field $F$ for study of smooth manifolds?
Roman "Advanced Linear Algebra". Isn't it an overkill for my goals? What do you think of this book?
Any other recommendations will be considered, of course. I would prefer modern books over likes of Hoffman-Kunze or Halmos, but if you know any old gems, feel free to tell. 
 A: The topics that you want to study use mostly the very essential ideas from Linear Algebra. Yes, over $\mathbb{R}$ and $\mathbb{C}$ is all you need. Since you seem to be a theoretically minded person with interest in geometry related subjects, I would recommend Gelfand's "Lectures on Linear Algebra". Strang's textbook is excellent, but probably not the style you are looking for.
My area of research is differential geometry (moving surfaces) and my favorite subjects to teach are Linear Algebra and Tensor Analysis. My advice would be to not treat Linear Algebra merely a means to studying the topics that you listed. Linear Algebra is the language and the framework for dealing with all branches of applied mathematics where physics, algebra and geometry meet. I would therefore recommend to study Linear Algebra thoroughly and, in fact, to give it several passes following a few different sources.
In case you want to learn my personal perspective on Linear Algebra and Differential Geometry, check out the Linear Algebra course on Lemma (http://lem.ma) and my Tensor Calculus videos on YouTube.
When Niels Henrik Abel was asked how he acquired his expertise, we replied "By studying the masters and not their pupils." Just one of my reactions to your desire to read a modern textbook.
