The term, modern algebra covers both linear algebra and non-linear algebra. It refers to a ranking of algebraic knowledge view relative to its history in terms of either model to solve problems or yet

in terms of the presentation of this knowledge.

Modern algebra grew out of classical algebra. Classical algebra has been learned today in middle and high school, and it developed irregularly over thousands of years. Modern algebra, also called abstract algebra, is a branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements.

Modern Algebra, was written in 1930 by the Dutch mathematician Bartel van der Waerden, and the subject has had a deep effect on almost every branch of mathematics.

Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.

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. The matrix and determinant are extremely useful tools of linear algebra. One central problem of linear algebra is the solution of the matrix equation

Ax= b

for x . While this can, in theory, be solved using a matrix inverse

x=A-1*b

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.

Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.

Based on what has been said before linear algebra can be considered as part of modern algebra.

thanks :Prof. Jorge Morales Pedraza , Prof. Viswanath Devan and Prof. Ahmed Charifi

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations .

Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation.

Modern algebra, also called abstract algebra, branch of mathematics conic , complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements.