There are essentially two prerequisites for studying linear algebra, besides the usual high school courses which include algebra, geometry and trigonometry (no, calculus is not a formal prerequisite). Those are that you must be used to proofs, and that you must have seen the basics of matrices and determinants.
For proofs, I recommend the following books in no particular order:
Velleman “How to prove it: a structured approach” http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
Bloch’s “Proofs and fundamentals: a first course in abstract mathematics” http://www.amazon.com/Proofs-Fundamentals-Abstract-Mathematics-Undergraduate/dp/1441971262
Hammack’s “Book of proof” http://www.people.vcu.edu/~rhammack/BookOfProof/
These books will teach you the fundamentals of proof based mathematics, and they will teach you the basic notations and assumptions of set theory. Both will be very necessary for algebra.
As for matrices and determinants, these should be covered in basic high school books. For example
Lang’s “Basic Mathematics” http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
Openstaxcollege’s Algebra https://openstaxcollege.org/textbooks/college-algebra
Khan Academy’s Linear algebra https://www.khanacademy.org/math/linear-algebra/matrix_transformations
Don’t worry, you don’t need to know matrices and determinants in full detail. Just knowing them for the 2×2 and 3×3 cases is already enough. In the linear algebra books which I’m going to mention, matrices and determinants will be treated in its full gory details. Also don’t worry if you don’t have a lot of intuition for matrices and determinants, this intuition will come with studying vector spaces. Right now you are merely expected to being able to compute stuff like inverses of matrices and solving systems of equations.
Linear algebra: Introduction
Proper linear algebra deals with vector spaces and linear transformations between them. It will rigorously develop the theory of nxn matrices and determinants and apply them to geometrical and algebraic contexts. As a book I highly recommend Friedberg, Insel, Spence’s “Linear algebra” http://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514 This book offers a very modern introduction to linear algebra. It deals with the following topics:
- Vector spaces (including linear dependence, subspaces, bases, dimension)
- Linear transformations and matrices (including rank, isomorphisms, change of basis matrices, dual spaces)
- Determinants including rigorous proofs of its properties
- Diagonalization (including eigenvalues, Markov chains and the Cayley-Hamilton theorem)
- Inner product spaces (including Gram-Schmidt process, the spectral theorem, singular value decomposition)
- Canonical forms (including the Jordan and the rational canonical form)
For an introduction to linear algebra, I highly prefer Friedberg, Insel, Spence over other books. Nevertheless, there are some other books on linear algebra that I really like, but that are perhaps not suited for an introduction. Let me list the other books that I really like.
Linear algebra done right by Axler
This is a great book with a lot of great proofs. I do not at all recommend it as a first introduction because of its treatment of determinants. Axler actively avoids determinants and only introduces them in a very abstract way in the last chapter. I think determinants are quite important in mathematics, so it deserves a better treatment than you’ll see in Axler’s book. A lot of proofs like the existence of eigenvectors are gems though.
Prerequisites: A course on linear algebra like Friedberg
Linear algebra done wrong by Treil
For a long time, this has been one of my favorite books on linear algebra. But I decided it is too terse for an introduction. It is very nice as a second book because it covers a lot of cool nontraditional topics like tensors. Also, it’s free!
Prerequisites: A course on linear algebra like Friedberg
Matrix Analysis and applied linear algebra by Meyer
This is a great book that focuses more on the applied numerical side of linear algebra. It contains a lot of insights and computational rules that cannot be found in other books of this level. Do you think that a low determinant means that a matrix is close to singular? This book will dispell you of these and similar notions.
3000 Solved problems in linear algebra by Lipschutz
Do you find yourself in need of more problems, this book is filled with them. A lot of computational, but also some proofy problems.
Linear and geometric algebra by Macdonald
This is a book that not only covers linear algebra, but also geometric algebra. Geometric algebra is a great theory which is very applicable in physics and computer science. In mathematics they are known as Clifford algebras. They form a neat unification of quaternions, forms, etc. It also presents a very neat way of seeing the determinant, something that introductory courses on linear algebra do not present.
Geometric algebra for computer science by Dorst, Fontijne, Mann
This is an alternative for MacDonald. Don’t be turned down by “computer science” in the title, this is a really nice book that offers a lot of neat intuition on geometric algebra. It covers a lot more of Geometric algebra than MacDonalds book.
Linear algebra via exterior products by Winitzki
This is a book that covers basis-free linear algebra. It makes extensive use of the wedge product, and not of usual matrix and vector computations. It is a nice companion to learning geometric algebra. It is very good as a sequel to the usual linear algebra books.
Prerequisites: A course in linear algebra like Friedberg
Advanced Linear Algebra by Roman
This book offers the most advanced view of linear algebra. It covers topics like modules, Hilbert spaces and even umbral calculus. Don’t take this book lightly though, you’ll need some more knowledge and mathematical maturity to be able to handle this book. Abstract algebra, calculus and analysis is a must. This book is a graduate book for a reason!
Prerequisites: Analysis, Abstract algebra and Linear algebra