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
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 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.
