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Algebra is a subject that has many areas of study. One of the areas of study in this branch of mathematics is linear algebra. So what is linear algebra?

It's a core unit in the heart of mathematical areas. In geometry, is used in the fundamentals if geometrical presentations such as rotations, lines, and planes. Another vital field in geometry that uses this field is functional analysis. In engineering mathematics, linear algebra is of great usage since it is good at computing natural phenomena's models such as the linear engineering systems. For the nonlinear system, It can also be used for approximation purpose.

History of linear algebra

The use of simultaneous linear equations can be traced back in the 15th century. These procedures are today known as the Gaussian elimination, which can be found in the ancient Chinese mathematics book called "The nine chapters of the mathematical art."

This field arose in 1637 in Europe as the systems of linear equations. It was introduced by Rene Descartes of geometry coordinates. The coordinates of geometry are today called Cartesian geometry. In Cartesian geometry, the linear equations are used to represent the planes and lines which are used to compute the intersections.

In 1693, Leibniz introduces the first systematic way to find the solutions of linear systems by the use of determinants. In 1750, there was a further improvement in this field by Gabriel Cramer who used the determinants to give clear answers to linear systems. This method of Gabriel Crammer is known as Cramer's rule. The Gauss later described the linear equation solving method that was termed as the elimination method which was the best advancement in the world of linear equations.

Hermann Grassmann later published a book on the "Theory of Extension" in 1844. This theory of extension includes topics which are today known as the Linear Algebra. Later in 1848, a well-known mathematician, James J. Sylvester, introduced what is today known as the linear matrix. The term matrix comes from the Latin word which means Womb.

Axioms in linear algebra

The linear algebra use axioms in the calculation of its equations. Some of these axioms are

  • Associativity of addition. This axiom is denoted as u+(v+w)=?(u+v)+w
  • Identity element of addition denoted as v+0=v for all the zero vectors
  • Commutativity of addition which is commonly denoted as u+v=v+u
  • Uses of Linear algebra

    It's widely used in related mathematical operations. Some of the uses are

    Functional analysis
    Functional analysis is widely used to study the function spaces. Function spaces are vector spaces that have additional structure. A good example of function space is the Hilbert Spaces. Therefore, the use of linear algebra in functional analysis is fundamental especially in the quantum mechanics which is widely known as wave functions.

    Scientific calculations and complex systems
    Linear algebra is used in the partial differential equations of all scientific computations. The differential equations are used to calculate the large matrices of linear systems such as the weather forecasting.


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