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# Linear Algebra

Linear algebra is 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 problems in mathematics, physics, and engineering. Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space over a field , and so on).

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

for . While this can, in theory, be solved using a matrix inverse

other techniques such as Gaussian elimination are numerically more robust.

In addition to being used to describe the study of linear sets of equations, the term "linear algebra" is also used to describe a particular type of algebra. In particular, a linear algebra over a field has the structure of a ring with all the usual axioms for an inner addition and an inner multiplication together with distributive laws, therefore giving it more structure than a ring. A linear algebra also admits an outer operation of multiplication by scalars (that are elements of the underlying field ). For example, the set of all linear transformations from a vector space to itself over a field forms a linear algebra over . Another example of a linear algebra is the set of all real square matrices over the field of the real numbers.

Abstract Algebra, Control Theory, Cramer's Rule, Determinant, Gaussian Elimination, Linear Transformation, Matrix, Vector Explore this topic in the MathWorld classroom

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## References

Axler, S. Linear Algebra Done Right, 2nd ed. New York: Springer-Verlag, 1997.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.BLAS. "BLAS (Basic Linear Algebra Subprograms)." http://www.netlib.org/blas/.Carlson, D.; Johnson, C. R.; Lay, D. C.; Porter, A. D.; Watkins, A. E.; and Watkins, W. (Eds.). Resources for Teaching Linear Algebra. Washington, DC: Math. Assoc. Amer., 1997.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.LAPACK. "LAPACK--Linear Algebra PACKage." http://www.netlib.org/lapack/.Lipschutz, S. Schaum's Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, 1991.Lumsdaine, J. and Siek, J. "The Matrix Template Library: Generic Components for High Performance Scientific Computing." http://www.lsc.nd.edu/research/mtl/.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.Petard, H. Problems in Linear Algebra, preliminary ed. New York: W.A. Benjamin, 1967.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.Weisstein, E. W. "Books about Linear Algebra." http://www.ericweisstein.com/encyclopedias/books/LinearAlgebra.html.Zhang, F. Matrix Theory: Basic Results and Techniques. New York: Springer-Verlag, 1999.

Linear Algebra

## Cite this as:

Weisstein, Eric W. "Linear Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LinearAlgebra.html