Linear algebra is a simple mathematical tool that is easy to use and is the necessary background for more complicated analytical methods. Linear algebra's main purpose is to set up and solve multiple problems simultaneously. In short, it is an extension of algebra to matrices. Its main application is in dealing with situations with large sets of data that must be entered into matrices. Through linear algebra, economists can analyze the matrices for important information, including relationships among economic variables.
Linear programming, which is partly based off linear algebra, is a quantitative method of analyzing systems under constraint. This type of analysis is especially important in economics, because economists must deal with problems in which people or organizations are limited by certain factors, including resource limitation, environmental constraint and legal restrictions. Linear programming differs from linear algebra in that the primary mathematical language in linear programming is inequalities instead of equations.
Nonlinear programming is an extension of linear programming to situations in which constraints are complex. This complexity makes the analysis more complex, and thus requires a new form of analysis. Nonlinear programming allows economists to examine situations in which constraints become nonlinear, including quadratic, logistic and exponential constraints. An example of a situation that uses nonlinear programming for analysis is one in which resources are being depleted at a more-than-linear rate, such as is the case with the ocean's fish.
Despite the name, game theory is a sincere analytical method in economics; though you can apply game theory to the study of games, the truth of the matter is that this analytical tool is most often used in economics. Using game theory, economists create game-like matrices to represent real-world economic situations. Economists then analyze these matrices to understand the trends and solutions to the situations. Game theory is integral to the study of decision-making, economic competition and marketing.