A rigorous book on a First Course in linear algebra I am just going to start linear algebra for my undergraduate course. And which is the best book available for linear algebra in terms of rigor and a book similar to calculus by Tom M Apostol (I like this book because of it's rigorous and clear ideas, the qualities which I want).
Note: I have browsed through other similar topics here and didn't found anything helpful.
 A: For a completely rigorous first account, I can recommend (based on my own experience from 1st and 2nd year undergrad) Friedberg–Insel–Spence's Introduction to Linear Algebra, though I seem to recall the account of determinants being a bit on the clumsy side. On the other hand, Axler's Linear Algebra Done Right has quite a good reputation.
Once you've learnt some linear algebra, though, Halmos's Finite-Dimensional Vector Spaces makes for an excellent second course; Roman's Advanced Linear Algebra, at least insofar as I've consulted it as a reference, strikes me as quite good as well.
A: If you've already read Apostol's Calculus book, I think you're ready for Strang's book called Linear Algebra and Its Applications. Strang's other book Introduction to Linear Algebra was mentioned above, but I think you should go right to the more serious book. 
I really can't overstate how good Linear Algebra and Its Applications is. It teaches you to think about linear algebra the right way. As you go on, you may need other books, but Strang should take you a long, long way. Here are some other options:
Roman: very good, better once you have had a rigorous introduction to abstract algebra (particularly finitely generated modules over PIDs)
Friedberg: Great exercises, OK chapters. I find the chapters difficult to read because they are so pedantic, using lengthy and cumbersome methods to prove things which are actually quite simple. Once you've read Strang's book, instead of reading Friedberg, I would just take a course on abstract algebra (including module theory) and then go back and read Roman (a large part of which you'll have covered in your algebra course, but some of it you won't have). But do the exercises in Friedberg!
Lax: This book is notable for its speed (duals and quotients are in the first 20 pages). The great part about Lax is he covers a bunch of things other linear algebra books don't, like matrix calculus. (For instance, if you have a differentiable function $A(t)$ taking as its values invertible matrices, what is the derivative of $A^{-1}(t)$?) This is really a great book, but I would never recommend it as a first book. One drawback: it has a few serious typos.
A: Check this book by Gilbert Strang 

"Introduction to Linear Algebra"

