Linear discriminant analysis is a part of the larger field of linear algebra. Linear algebra does not deal exclusively with lines, but rather with a set of variables which are raised only to the zeroth or first power. Linear algebra has been around since the late 1600s when Leibnitz, the developer of calculus, used rudimentary methods to study the coefficients of linear equations. It was, however, the advent of the computer that lifted the field into prominence.
Linear discriminant analysis is closely related to linear regression. Discriminant analysis, rather than dealing exclusively with the variables related to the data, deals with the groups or the classes to which the data applies. It also deals with the variance among the classes.
You can use discriminant analysis to predict which companies may go bankrupt based on financial variables. The classes being those who go bankrupt and those who don't. You can use discriminant analysis in marketing to separate customers into different types and classes base on data collected in surveys. Facial recognition based on pixel grouping is another application. Psychologists and sociologists can also use discriminant analysis.