Linear Algebra

Course Outline

Linear Algebra

LINEAR ALGEBRA

  • Definition of a vector space (over R or C), subspaces, the space spanned by a subset. Linear independence, bases, dimension. Direct sums and complementary subspaces. Quotient spaces. [3]
  • Linear maps, isomorphisms. Relation between rank and nullity. The space of linear maps from U to V , representation by matrices. Change of basis. Row rank and column rank. [4]
  • Determinant and trace of a square matrix. Determinant of a product of two matrices and of the inverse matrix. Determinant of an endomorphism. The adjugate matrix. [3]
  • Eigenvalues and eigenvectors. Diagonal and triangular forms. Characteristic and minimal polynomials. Cayley–Hamilton Theorem over C. Algebraic and geometric multiplicity of eigenvalues. Statement and illustration of Jordan normal form. [4]
  • Dual of a finite-dimensional vector space, dual bases and maps. Matrix representation, rank and determinant of dual map [2]
  • Bilinear forms. Matrix representation, change of basis. Symmetric forms and their link with quadratic forms. Diagonalisation of quadratic forms. Law of inertia, classification by rank and signature. Complex Hermitian forms. [4]
  • Inner product spaces, orthonormal sets, orthogonal projection, V = W ⊕ W⊥. Gram–Schmidt orthogonalisation. Adjoints. Diagonalisation of Hermitian matrices. Orthogonality of eigenvectors and properties of eigenvalues. [4]

Appropriate books

  • C.W. Curtis Linear Algebra: an introductory approach. Springer 1984
  • P.R. Halmos Finite-dimensional vector spaces. Springer 1974
  • K. Hoffman and R. Kunze Linear Algebra. Prentice-Hall 1971

References:

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

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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