Introduction to 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.