If you would follow the road to linear algebra here are some trustworthy signposts.

Generalist

These books develop the subject with minimal prerequisites. They cover a broad range of theory and selected applications.

Axler, S. J. (2014). Linear algebra done right. New York: Springer-Verlag.

Curtis, C. W. (1984). Linear algebra: An introductory approach. New York: Springer-Verlag.

Greub, W. H. (1975). Linear algebra. New York: Springer-Verlag.

Halmos, P. R. (1958). Finite-dimensional vector spaces. Princeton, N.J: Van Nostrand.

Herstein, I. N., & Winter, D. J. (1988). A primer on linear algebra. New York: Macmillan.

Hoffman, K., & Kunze, R. A. (1971). Linear algebra. Englewood Cliffs, N.J: Prentice-Hall.

Katznelson, Y., & Katznelson, Y. R. (2008). A (terse) introduction to linear algebra. Providence, R.I: American Mathematical Society.

Lax, P. D., & Lax, P. D. (2007). Linear algebra and its applications. Hoboken, N.J: Wiley-Interscience.

Historical

All mathematics is a work in progress and we should never take its current definitions as sacred. Learn the evolution of linear algebra and see how different formulations battled it out.

Crowe, M. J. (1994). A history of vector analysis: The evolution of the idea of a vectorial system. Dover Pub.

Grassmann, H. (1995). A new branch of mathematics: The “Ausdehnungslehre” of 1844 and other works. Chicago: Open Court.

Theoretical

These books develop the subject rigorously, often on generalized assumptions.

Aluffi, P. (2009). Algebra: Chapter 0. Providence, R.I: American Mathematical Society.

Blyth, T. S. (1990). Module theory: An approach to linear algebra. Oxford [England: Clarendon Press.

Bourbaki, N. (1989). Algebra I. Berlin: Springer-Verlag.

Brown, W. C. (1988). A second course in linear algebra. New York: Wiley.

Curtis, M. L., & Place, P. (1990). Abstract linear algebra. New York: Springer-Verlag.

Golan, J. S. (2012). The linear algebra a beginning graduate student ought to know. Dordrecht: Springer.

Jacobson, N. (1951). Lectures in abstract algebra: Linear algebra. 2 eks. New York, Van Nostrand Reinhold.

Lang, S. (1987). Linear algebra. New York: Springer-Verlag.

Roman, S. (2008). Advanced linear algebra. New York: Springer.

Shakarchi, R., & Lang, S. (1996). Solutions manual for Lang’s Linear algebra. New York: Springer.

Valenza, R. J. (1993). Linear algebra: An introduction to abstract mathematics. New York: Springer-Verlag.

Weintraub, S. H., & Mathematical Association of America. (2011). A guide to advanced linear algebra. Washington, DC: Mathematical Association of America.

Numerical

These describe matrix forms and efficient algorithms for getting numerical answers.

Golub, G. H., & Van, L. C. F. (2013). Matrix computations.

Herstein, I. N., & Winter, D. J. (1988). Matrix theory and linear algebra. New York: Macmillan.

Horn, R. A., & Johnson, C. R. (2013). Matrix analysis, second edition. Cambridge: Cambridge University Press.

Trefethen, L. N., & Bau, D. (1997). Numerical linear algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics.

Practice, practice

While all math books provide exercises, these books are comprised entirely of them, along with hints and solutions.

Blyth, T. S., & Robertson, E. F. (1984). Algebra through practice: A collection of problems in algebra, with solutions. Cambridge: Cambridge University Press.

Halmos, P. R. (1995). Linear algebra problem book. Washington, DC: Mathematical Association of America.

Lipschutz, S. (1988). Three thousand solved problems in linear algebra. New York: McGraw-Hill.

Prasolov, V. V., & Ivanov, S. (1994). Problems and theorems in linear algebra. Providence, R.I: American Mathematical Society.

Zhang, F. (1996). Linear algebra: Challenging problems for students. Baltimore: Johns Hopkins University Press.

Extended

Directions for further study.

Greub, W. H. (1978). Multilinear algebra. New York: Springer-Verlag.

Rudin, W. (1991). Functional analysis. New York: McGraw-Hill.

Schaefer, H. H., & Wolff, M. P. (1999). Topological vector spaces. S.l.: Springer.

Weinreich, G. (1998). Geometrical vectors. Chicago: University of Chicago Press.

Weird Russian

Supposedly using outdated notation but packed with wisdom.

Akivis, M. A., Golʹdberg, V. V., & Silverman, R. A. (1977). An introduction to linear algebra and tensors. New York: Dover Publications.

Gelʹfand, I. M. (1989). Lectures on linear algebra. New York: Dover Publications.

Shilov, G. E. (1977). Linear algebra. New York: Dover Publications.

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