I came to know about Gilbert Strang's two books, "Introduction to Linear Algebra" and "Linear Algebra and its Applications". The first is the one used as the text in the 18.06 linear algebra MIT course. I have almost all the videos of 18.06 which deal with basics linear algebra (like left out the portions of Markov matrices, differential equations, Fourier series, etc). Though Gilbert Strang's videos are awesome and highly intuitive, it does not define many things, from which I have encountered questions like: Block determinants, rank, nullity theorem, the general form of the characteristic equation (using the concept that rth co-efficient is the sum of the principal minors of order r), principal minors, and many other small pieces which can be used to answer MCQ question in competitive examinations.
Though the second book, has concepts like rank, nullity, etc, but even there, many concepts which I need are missing, and it is more inclined towards application stuff.
Please can anyone recommend me an intuitive yet comprehensive and easily readable (student-friendly) book on Linear Algebra which do not focus much on applications, just basics