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Linear algebra is the branch of mathematics concerning linear equations such as: ... 1989), 3,000 Solved Problems in Linear Algebra, McGraw–Hill, ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Representation theory transforms abstract algebra groups into things like simpler matrices. The field’s founder left a list of 43 problems for others to study, iterate on, and prove.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
There are algorithms to solve all the problems addressed in this article over the integers. In other words, linear algebra is effective over the integers; see Linear Diophantine system for details. More generally, linear algebra is effective on a principal ideal domain if there are algorithms for addition, subtraction and multiplication, and
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
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