Linear Algebra

The study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of Rotations in space, Least Squares Fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other other problems in mathematics, physics, and engineering.

The Matrix and Determinant are extremely useful tools of linear algebra. One central problem of linear algebra is the solution of the matrix equation

$${\hbox{\sf A}}{\bf x}={\bf b}$$

for x. While this can, in theory, be solved using a Matrix Inverse

$${\bf x}={\hbox{\sf A}}^{-1}{\bf b},$$

other techniques such as Gaussian Elimination are numerically more robust.

See also Control Theory, Cramer's Rule, Determinant, Gaussian Elimination, Linear Transformation, Matrix, Vector

References

Linear Algebra

Ayres, F. Jr. Theory and Problems of Matrices. New York: Schaum, 1962.

Banchoff, T. and Wermer, J. Linear Algebra Through Geometry, 2nd ed. New York: Springer-Verlag, 1992.

Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New York: McGraw-Hill, 1970.

Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, 1958.

Golub, G. and van Loan, C. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.

Halmos, P. R. Linear Algebra Problem Book. Providence, RI: Math. Assoc. Amer., 1995.

Lang, S. Introduction to Linear Algebra, 2nd ed. New York: Springer-Verlag, 1997.

Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, 1988.

Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, 1992.

Marcus, M. Matrices and Matlab: A Tutorial. Englewood Cliffs, NJ: Prentice-Hall, 1993.

Mirsky, L. An Introduction to Linear Algebra. New York: Dover, 1990.

Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960.

Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, 1990.

Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.

Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993.

Strang, G. and Borre, K. Linear Algebra, Geodesy, & GPS. Wellesley, MA: Wellesley-Cambridge Press, 1997.