Linear algebra is the study of sets of linear 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
In addition to being used to describe the study of sets of linear 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 realsquare matrices
over the field of the real numbers.
over a field 
, and so on).
.
While this can, in theory, be solved using a matrix
inverse
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.
