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.
SEE ALSO: Abstract Algebra,
Control Theory,
Cramer's Rule,
Determinant,
Gaussian Elimination,
Linear
Transformation,
Matrix,
Vector
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Referenced on Wolfram|Alpha:
Linear Algebra
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